Finite Group Databasebeta

← C2×#22561005 #30859413

Subgroups — C2×#22561005 #30859413

Order 17152.

2094 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D67C211yesmaximal, characteristic
2C22×Dic67⋊C8C211yesmaximal, characteristic, complemented
3C22×C22⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
4C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
5C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
6C22×#14523953C211yesmaximal, characteristic, complemented
7#22561005C218yesmaximal, direct product factor
8C22×C22⋊C8×C67C211yesmaximal, characteristic, complemented
9C23×C22⋊D67C411yescharacteristic
10C23×C4×D67C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C23×C67⋊C8C2211yescharacteristic
13C2×C4×C22⋊D67—24no—
14C2×C4×C22⋊D67—24no—
15C2×C4×C22⋊D67C414yes—
16C2×C4×C22⋊D67C2214yes—
17C22×Dic67⋊C4C411yescharacteristic
18C22×C22⋊Dic67C411yescharacteristic
19C22×(C2×D67)⋊C4C411yescharacteristic
20C22×C4×Dic67C2211yescharacteristic
21C2×Dic67⋊C8—24no—
22C2×Dic67⋊C8C2214yescomplemented
23C2×C22⋊(C67⋊C8)—24no—
24C2×C22⋊(C67⋊C8)C2214yescomplemented
25C2×(C2×D67)⋊C8—24no—
26C2×(C2×D67)⋊C8—24no—
27C2×(C2×D67)⋊C8C2214yescomplemented
28C2×(C2×D67)⋊C8C2214yescomplemented
29C2×#14523953—24no—
30C2×#14523953C2214yescomplemented
31C24×C4×C67C2211yescharacteristic
32C23×C8×C67C2211yescharacteristic
33C23×C8×C67C2211yescharacteristic
34C2×C22⋊C8×C67—24no—
35C2×C22⋊C8×C67C2214yescomplemented
36C24×D67C2×C411yescharacteristic
37C22×C22⋊D67—24no—
38C22×C22⋊D67—24no—
39C22×C22⋊D67—24no—
40C22×C22⋊D67C2×C414yes—
41C23×Dic67C2×C411yescharacteristic
42C23×Dic67C2×C411yescharacteristic
43C23×Dic67C2×C411yescharacteristic
44C22×C4×D67—22no—
45C22×C4×D67—22no—
46C22×C4×D67—22no—
47C22×C4×D67—22no—
48C22×C4×D67—22no—
49C22×C4×D67—22no—
50C22×C4×D67C2×C412yes—
51C22×C4×D67C2312yes—
52C22×C67⋊C8—22no—
53C22×C67⋊C8—22no—
54C22×C67⋊C8—22no—
55C22×C67⋊C8—22no—
56C22×C67⋊C8—22no—
57C22×C67⋊C8D414yescomplemented
58C22×C67⋊C8C2312yes—
59C22×C67⋊C8C2312yes—
60C4×C22⋊D67—216no—
61C4×C22⋊D67—28no—
62C4×C22⋊D67—28no—
63C2×Dic67⋊C4—24no—
64C2×Dic67⋊C4C2×C414yes—
65C2×C22⋊Dic67—24no—
66C2×C22⋊Dic67C2×C414yes—
67C2×(C2×D67)⋊C4—24no—
68C2×(C2×D67)⋊C4C2×C414yes—
69C2×C4×Dic67C2×C412yes—
70C2×C4×Dic67D414yes—
71C2×C4×Dic67D414yes—
72C2×C4×Dic67C2312yes—
73Dic67⋊C8—28no—
74C22⋊(C67⋊C8)—28no—
75(C2×D67)⋊C8—28no—
76(C2×D67)⋊C8—28no—
77#14523953—28no—
78C25×C67C2×C411yescharacteristic
79C23×C4×C67—22no—
80C23×C4×C67—22no—
81C23×C4×C67—22no—
82C23×C4×C67—22no—
83C23×C4×C67—22no—
84C23×C4×C67—22no—
85C23×C4×C67C2×C411yescharacteristic
86C23×C4×C67C2×C412yes—
87C23×C4×C67C2312yes—
88C23×C4×C67C2311yescharacteristic
89C22×C8×C67—22no—
90C22×C8×C67—22no—
91C22×C8×C67—22no—
92C22×C8×C67—22no—
93C22×C8×C67—22no—
94C22×C8×C67D414yescomplemented
95C22×C8×C67C2312yes—
96C22×C8×C67C2312yes—
97C22⋊C8×C67—28no—
98C23×D67—22no—
99C23×D67—22no—
100C23×D67—22no—
101C23×D67—22no—
102C23×D67—22no—
103C23×D67—22no—
104C23×D67M1614yes—
105C23×D67C22×C412yes—
106C2×C22⋊D67—232no—
107C2×C22⋊D67—28no—
108C2×C22⋊D67—28no—
109C2×C22⋊D67—28no—
110C22×Dic67—22no—
111C22×Dic67—22no—
112C22×Dic67—22no—
113C22×Dic67—22no—
114C22×Dic67—22no—
115C22×Dic67—22no—
116C22×Dic67—22no—
117C22×Dic67C22⋊C414yes—
118C22×Dic67C22⋊C414yes—
119C22×Dic67C22×C412yes—
120C22×Dic67C22×C412yes—
121C22×Dic67C22×C412yes—
122C2×C4×D67—24no—
123C2×C4×D67—24no—
124C2×C4×D67—24no—
125C2×C4×D67—22no—
126C2×C4×D67—22no—
127C2×C4×D67—22no—
128C2×C4×D67—22no—
129C2×C4×D67—22no—
130C2×C4×D67—22no—
131C2×C4×D67—44no—
132C2×C4×D67—44no—
133C2×C4×D67—44no—
134C2×C4×D67—44no—
135C2×C67⋊C8—28no—
136C2×C67⋊C8—22no—
137C2×C67⋊C8—22no—
138C2×C67⋊C8—22no—
139C2×C67⋊C8—22no—
140C2×C67⋊C8—22no—
141C2×C67⋊C8—44no—
142C2×C67⋊C8C2×D414yescomplemented
143Dic67⋊C4—28no—
144C22⋊Dic67—28no—
145(C2×D67)⋊C4—28no—
146C4×Dic67C22⋊C418yes—
147C4×Dic67C2×D414yes—
148C4×Dic67C2×D414yes—
149C24×C67—22no—
150C24×C67—22no—
151C24×C67—22no—
152C24×C67—22no—
153C24×C67—22no—
154C24×C67—22no—
155C24×C67M1614yes—
156C24×C67C22×C412yes—
157C24×C67C22×C411yescharacteristic
158C22×C4×C67—22no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—24no—
162C22×C4×C67—24no—
163C22×C4×C67—24no—
164C22×C4×C67—22no—
165C22×C4×C67—22no—
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—22no—
169C22×C4×C67—22no—
170C22×C4×C67—44no—
171C22×C4×C67—44no—
172C22×C4×C67—44no—
173C22×C4×C67—44no—
174C22×C4×C67C22×C412yes—
175C22×C4×C67C22×C411yescharacteristic
176C22×C4×C67C2×D412yes—
177C22×C4×C67C2×D412yes—
178C22×C4×C67C2×D412yes—
179C22×C4×C67C2×D412yes—
180C22×C4×C67#4112yes—
181C22×C4×C67#4112yes—
182C22×C4×C67C2411yescharacteristic
183C2×C8×C67—28no—
184C2×C8×C67—22no—
185C2×C8×C67—22no—
186C2×C8×C67—22no—
187C2×C8×C67—22no—
188C2×C8×C67—22no—
189C2×C8×C67—44no—
190C2×C8×C67C2×D414yescomplemented
191C23×C67—22no—
192C23×C67—22no—
193C23×C67—22no—
194C23×C67—22no—
195C23×C67—22no—
196C23×C67—22no—
197C23×C67—28no—
198C23×C67—28no—
199C23×C67—28no—
200C23×C67—44no—
201C23×C67—44no—
202C23×C67—44no—
203C23×C67—44no—
204C23×C67C2×C22⋊C412yes—
205C23×C67C4×D412yes—
206C23×C67C4×D412yes—
207C23×C67C2×M1612yes—
208C23×C67C2×M1614yes—
209C23×C67C2×M1612yes—
210C23×C67#13112yes—
211C23×C67#13112yes—
212C23×C67C23×C411yescharacteristic
213C22×D67—22no—
214C22×D67—22no—
215C22×D67—22no—
216C22×D67—22no—
217C22×D67—22no—
218C22×D67—22no—
219C22×D67—28no—
220C22×D67—28no—
221C22×D67—28no—
222C22×D67—44no—
223C22×D67—44no—
224C22×D67—44no—
225C22×D67—44no—
226C22×D67C2×M1614yes—
227C2×C4×C67—22no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—44no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—48no—
236C2×C4×C67C2×C22⋊C412yes—
237C2×C4×C67C4×D412yes—
238C2×C4×C67C4×D412yes—
239C2×C4×C67#12114yes—
240C2×C4×C67#12114yes—
241C2×C4×C67#12414yes—
242C2×C4×C67#12714yes—
243C2×C4×C67C22×D411yescharacteristic
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C2×#4111yescharacteristic
248C2×C4×C67C2×#4111yescharacteristic
249C22⋊D67—232no—
250C2×Dic67—22no—
251C2×Dic67—22no—
252C2×Dic67—22no—
253C2×Dic67—22no—
254C2×Dic67—28no—
255C2×Dic67—28no—
256C2×Dic67—22no—
257C2×Dic67—22no—
258C2×Dic67—22no—
259C2×Dic67—44no—
260C2×Dic67C2×C22⋊C414yes—
261C2×Dic67C2×C22⋊C414yes—
262C4×D67—42no—
263C4×D67—42no—
264C4×D67—42no—
265C4×D67—42no—
266C4×D67—48no—
267C8×C67—24no—
268C8×C67—42no—
269C67⋊C8—24no—
270C67⋊C8—42no—
271C22×C67—24no—
272C22×C67—24no—
273C22×C67—24no—
274C22×C67—42no—
275C22×C67—42no—
276C22×C67—42no—
277C22×C67—416no—
278C22×C67—42no—
279C22×C67#39014yes—
280C22×C67#40714yes—
281C22×C67#40814yes—
282C22×C67#43514yes—
283C22×C67#43514yes—
284C22×C67#43514yes—
285C22×C67#43514yes—
286C22×C67C22×C22⋊C411yescharacteristic
287C22×C67C2×C4×D411yescharacteristic
288C22×C67C2×C4×D411yescharacteristic
289C22×C67C22×M1611yescharacteristic
290C22×C67C22×M1611yescharacteristic
291C22×C67C2×#13111yescharacteristic
292C22×C67C2×#13111yescharacteristic
293C2×D67—24no—
294C2×D67—24no—
295C2×D67—24no—
296C2×D67—42no—
297C2×D67—42no—
298C2×D67—42no—
299C2×D67—416no—
300C2×D67—42no—
301C4×C67—42no—
302C4×C67#39014yes—
303C4×C67C2×#12111yescharacteristic
304C4×C67C2×#12111yescharacteristic
305C4×C67C2×#12411yescharacteristic
306C4×C67C2×#12711yescharacteristic
307Dic67—24no—
308Dic67—24no—
309Dic67—42no—
310C2×#1955—671nomaximal, Sylow
311C2×C67—44no—
312C2×C67#195518yescomplemented
313C2×C67C2×#39011yescharacteristic
314C2×C67C2×#40711yescharacteristic
315C2×C67C2×#40811yescharacteristic
316C2×C67C2×#43511yescharacteristic
317C2×C67C2×#43511yescharacteristic
318C2×C67C2×#43511yescharacteristic
319C2×C67C2×#43511yescharacteristic
320D67—44no—
321C22×C4×D4—671no—
322C22×C4⋊C8—671no—
323C22×C22⋊C8—671no—
324C22×C22⋊C8—671no—
325C22×C22⋊C8—671no—
326C22×C22⋊C8D6711yescharacteristic, complemented
327C22×#97—671no—
328#1955—678no—
329C67C2×#195511yescharacteristic, complemented, Sylow
330C23×D4—671no—
331C24×C4—671no—
332C24×C4C2×D6711yescharacteristic
333C23×C8—671no—
334C23×C8—671no—
335C23×C8C2×D6711yescharacteristic
336C23×C8C2×D6711yescharacteristic
337C2×C4×D4—674no—
338C2×C4×D4—674no—
339C2×C4×D4—1344no—
340C2×C4×D4—1344no—
341C22×C4⋊C4—671no—
342C22×C22⋊C4—671no—
343C22×C22⋊C4—671no—
344C22×C42—671no—
345C2×C4⋊C8—674no—
346C2×C4⋊C8—1344no—
347C2×C22⋊C8—24no—
348C2×C22⋊C8—674no—
349C2×C22⋊C8—674no—
350C2×C22⋊C8—674no—
351C2×C22⋊C8—1344no—
352C2×C22⋊C8—1344no—
353C2×C22⋊C8—1344no—
354C2×C22⋊C8C2×D6714yescomplemented
355C2×#97—674no—
356C2×#97—1344no—
357C25—671no—
358C25C4×D6711yescharacteristic
359C22×D4—674no—
360C22×D4—1344no—
361C22×D4—1344no—
362C22×D4—1344no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—671no—
370C23×C4—671no—
371C23×C4—671no—
372C23×C4—672no—
373C23×C4—672no—
374C23×C4—1342no—
375C23×C4—1342no—
376C23×C4—1342no—
377C23×C4—1342no—
378C23×C4—1342no—
379C23×C4—1342no—
380C23×C4C4×D6711yescharacteristic
381C23×C4C4×D6712yes—
382C23×C4C22×D6712yes—
383C23×C4C22×D6711yescharacteristic
384C22×C8—22no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—674no—
390C22×C8—672no—
391C22×C8—672no—
392C22×C8—1342no—
393C22×C8—1342no—
394C22×C8—1342no—
395C22×C8—1342no—
396C22×C8—1342no—
397C22×C8C22⋊D6714yescomplemented
398C22×C8C22×D6712yes—
399C22×C8C22×D6712yes—
400C4×D4—13416no—
401C4×D4—1348no—
402C4×D4—1348no—
403C2×C4⋊C4—674no—
404C2×C4⋊C4—1344no—
405C2×C22⋊C4—674no—
406C2×C22⋊C4—674no—
407C2×C22⋊C4—1344no—
408C2×C22⋊C4—1344no—
409C2×C42—672no—
410C2×C42—674no—
411C2×C42—674no—
412C2×C42—672no—
413C4⋊C8—1348no—
414C22⋊C8—28no—
415C22⋊C8—1348no—
416C22⋊C8—1348no—
417C22⋊C8—1348no—
418#97—1348no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—674no—
426C24—672no—
427C24—1342no—
428C24—1342no—
429C24—1342no—
430C24—1342no—
431C24—1342no—
432C24—1342no—
433C24#1176161114yes—
434C24C2×C4×D6712yes—
435C24C2×C4×D6711yescharacteristic
436C2×D4—1348no—
437C2×D4—1348no—
438C2×D4—13432no—
439C2×D4—1348no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—2684no—
457C22×C4—2684no—
458C22×C4—2684no—
459C22×C4—2684no—
460C22×C4—674no—
461C22×C4—674no—
462C22×C4—672no—
463C22×C4—672no—
464C22×C4—672no—
465C22×C4—1342no—
466C22×C4—1342no—
467C22×C4—1342no—
468C22×C4—1344no—
469C22×C4—1344no—
470C22×C4—1344no—
471C22×C4—1342no—
472C22×C4—1342no—
473C22×C4—1342no—
474C22×C4—1342no—
475C22×C4—1342no—
476C22×C4—1342no—
477C22×C4—1342no—
478C22×C4—1342no—
479C22×C4—1342no—
480C22×C4—1342no—
481C22×C4C2×C4×D6712yes—
482C22×C4C2×C4×D6711yescharacteristic
483C22×C4D4×D6712yes—
484C22×C4D4×D6712yes—
485C22×C4D4×D6712yes—
486C22×C4C2×C22⋊D6712yes—
487C22×C4#1176162712yes—
488C22×C4#1176162812yes—
489C22×C4C23×D6711yescharacteristic
490C2×C8—28no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—2684no—
498C2×C8—674no—
499C2×C8—1342no—
500C2×C8—1342no—
501C2×C8—1342no—
502C2×C8—1348no—
503C2×C8—1342no—
504C2×C8—1342no—
505C2×C8C2×C22⋊D6714yescomplemented
506C4⋊C4—1348no—
507C22⋊C4—1348no—
508C22⋊C4—1348no—
509C42—678no—
510C42—674no—
511C42—674no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—28no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—28no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—2684no—
526C23—2684no—
527C23—2684no—
528C23—2684no—
529C23—674no—
530C23—1342no—
531C23—1342no—
532C23—1342no—
533C23—1348no—
534C23—1342no—
535C23—1342no—
536C23—1342no—
537C23—1348no—
538C23—1348no—
539C23C22⋊C4×D6712yes—
540C23(C22⋊D67)⋊C412yes—
541C23C4×C22⋊D6712yes—
542C23M16×D6712yes—
543C23C2×#1176161114yes—
544C23C2×#1176161112yes—
545C23#1452404912yes—
546C23#1452405112yes—
547C23C22×C4×D6711yescharacteristic
548D4—13432no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—2684no—
559C2×C4—2688no—
560C2×C4—2682no—
561C2×C4—2682no—
562C2×C4—2682no—
563C2×C4—2682no—
564C2×C4—674no—
565C2×C4—674no—
566C2×C4—1342no—
567C2×C4—1342no—
568C2×C4—1342no—
569C2×C4—1348no—
570C2×C4—1348no—
571C2×C4—1342no—
572C2×C4—1342no—
573C2×C4—1342no—
574C2×C4—1342no—
575C2×C4C22⋊C4×D6712yes—
576C2×C4(C22⋊D67)⋊C412yes—
577C2×C4C4×C22⋊D6712yes—
578C2×C4#1452401514yes—
579C2×C4#1452401614yes—
580C2×C4#1452402914yes—
581C2×C4#1452403914yes—
582C2×C4C2×D4×D6711yescharacteristic
583C2×C4C2×D4×D6711yescharacteristic
584C2×C4C2×D4×D6711yescharacteristic
585C2×C4C22×C22⋊D6711yescharacteristic
586C2×C4C2×#1176162711yescharacteristic
587C2×C4C2×#1176162811yescharacteristic
588C8—24no—
589C8—42no—
590C8—2682no—
591C8—1344no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—416no—
600C22—2682no—
601C22—2682no—
602C22—2682no—
603C22—2682no—
604C22—26816no—
605C22—1344no—
606C22—1344no—
607C22—1344no—
608C22#1686329614yes—
609C22#1686335814yes—
610C22#1686336114yes—
611C22#1686346414yes—
612C22#1686346514yes—
613C22#1686346714yes—
614C22#1686347014yes—
615C22C2×C22⋊C4×D6711yescharacteristic
616C22C2×(C22⋊D67)⋊C411yescharacteristic
617C22C2×C4×C22⋊D6711yescharacteristic
618C22C2×M16×D6711yescharacteristic
619C22C22×#1176161111yescharacteristic
620C22C2×#1452404911yescharacteristic
621C22C2×#1452405111yescharacteristic
622C4—42no—
623C4—2682no—
624C4—1344no—
625C4—1344no—
626C4#1686329614yes—
627C4C2×#1452401511yescharacteristic
628C4C2×#1452401611yescharacteristic
629C4C2×#1452402911yescharacteristic
630C4C2×#1452403911yescharacteristic
631C2—44no—
632C2—2684no—
633C2#2256100518yesdirect product factor
634C2C2×#1686329611yescharacteristic
635C2C2×#1686335811yescharacteristic
636C2C2×#1686336111yescharacteristic
637C2C2×#1686346411yescharacteristic
638C2C2×#1686346511yescharacteristic
639C2C2×#1686346711yescharacteristic
640C2C2×#1686347011yescharacteristic
subgroupG/H
centerC23×C4C22×D67
commutatorC23×C67C23×C4
FrattiniC22×C4C23×D67
FittingC22×C22⋊C8×C67C2
socleC24×C67C22×C4
prime radicalC23×C22⋊D67C4
cosocleC22×C4×C67C24
hypercenterC22×C22⋊C8D67
hypocenterC67C2×#1955
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23×C4 ◁ C22×C22⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×Dic67⋊C8C2semidirect
3. C22×C22⋊(C67⋊C8)C2semidirect
4. C22×(C2×D67)⋊C8C2semidirect
5. C22×(C2×D67)⋊C8C2semidirect
6. C22×#14523953C2semidirect
7. #22561005C2direct (N × H)
8. C22×C22⋊C8×C67C2semidirect
22. C2×Dic67⋊C8C22semidirect
24. C2×C22⋊(C67⋊C8)C22semidirect
27. C2×(C2×D67)⋊C8C22semidirect
28. C2×(C2×D67)⋊C8C22semidirect
30. C2×#14523953C22semidirect
35. C2×C22⋊C8×C67C22semidirect
57. C22×C67⋊C8D4semidirect
94. C22×C8×C67D4semidirect
142. C2×C67⋊C8C2×D4semidirect
190. C2×C8×C67C2×D4semidirect
312. C2×C67#1955semidirect
326. C22×C22⋊C8D67semidirect
329. C67C2×#1955semidirect
354. C2×C22⋊C8C2×D67semidirect
397. C22×C8C22⋊D67semidirect
505. C2×C8C2×C22⋊D67semidirect
633. C2#22561005direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4×C22⋊D672
C22×Dic67⋊C82
C22×C22⋊(C67⋊C8)2
C22×(C2×D67)⋊C82
C22×#145239532
#225610052
C22×C22⋊C8×C672
C2×#195567