Finite Group Databasebeta

← #30777409

Subgroups — #30777409

Order 17152.

1778 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863293C212yesmaximal
2C2×C4×Dic67⋊C4C211yesmaximal, characteristic
3C2×C4×C4⋊Dic67C211yesmaximal, characteristic
4C2×#16863247C212yesmaximal
5C2×#16863247C211yesmaximal, characteristic
6#22558443C212yesmaximal
7#22558440C212yesmaximal
8Dic67⋊#95C211yesmaximal, characteristic
9#22558433C211yesmaximal, characteristic
10#95×Dic67C211yesmaximal, characteristic
11C67×#1434C211yesmaximal, characteristic
12C22×C4⋊Dic67C2212yes—
13C22×Dic67⋊C4C2212yes—
14C22×Dic67⋊C4C2212yes—
15C22×Dic67⋊C4C2211yescharacteristic
16C22×Dic67⋊C4C2211yescharacteristic
17C22×C4×Dic67C2212yes—
18C22×C4×Dic67C2212yes—
19C22×C4×Dic67C2212yes—
20C22×C4×Dic67C2211yescharacteristic
21C22×C4×Dic67C2211yescharacteristic
22#16863293C4116yescomplemented
23C4×Dic67⋊C4C418yescomplemented
24C4×C4⋊Dic67—24no—
25#16863247—28no—
26#16863247—24no—
27C2×C67⋊#95C2212yes—
28C2×C67⋊#95C2211yescharacteristic
29C2×C67⋊#95C2212yes—
30C2×C67⋊#95C2212yes—
31C2×C67⋊#95C2212yes—
32C2×C67⋊#95C2211yescharacteristic
33C2×C67⋊#95C2211yescharacteristic
34C2×C67⋊#95C2211yescharacteristic
35C22×C42×C67C2211yescharacteristic
36C2×#95×C67C2211yescharacteristic
37C2×#95×C67C2211yescharacteristic
38C2×#95×C67C2212yes—
39C2×#95×C67C2212yes—
40C23×Dic67C2311yescharacteristic
41C23×Dic67C2312yes—
42C23×Dic67C2311yescharacteristic
43C23×Dic67C2311yescharacteristic
44C23×Dic67C2312yes—
45C23×Dic67C2311yescharacteristic
46C2×C4⋊Dic67—24no—
47C2×C4⋊Dic67—24no—
48C2×C4⋊Dic67C2×C418yes—
49C2×Dic67⋊C4—24no—
50C2×Dic67⋊C4—24no—
51C2×Dic67⋊C4—24no—
52C2×Dic67⋊C4—24no—
53C2×Dic67⋊C4C2×C418yes—
54C2×Dic67⋊C4C2×C418yes—
55C2×Dic67⋊C4C2×C414yes—
56C2×Dic67⋊C4C2×C418yes—
57C2×Dic67⋊C4C2×C414yes—
58C2×Dic67⋊C4D414yes—
59C2×Dic67⋊C4Q814yes—
60C2×C4×Dic67—24no—
61C2×C4×Dic67—24no—
62C2×C4×Dic67—24no—
63C2×C4×Dic67—24no—
64C2×C4×Dic67—24no—
65C2×C4×Dic67—24no—
66C2×C4×Dic67—22no—
67C2×C4×Dic67—24no—
68C2×C4×Dic67—22no—
69C2×C4×Dic67—42no—
70C2×C4×Dic67—42no—
71C2×C4×Dic67C2×C414yes—
72C2×C4×Dic67C2×C414yes—
73C2×C4×Dic67C2×C418yes—
74C67⋊#95—24no—
75C67⋊#95—24no—
76C67⋊#95—22no—
77C67⋊#95—42no—
78C67⋊#95C2×C414yes—
79C67⋊#95C2×C414yes—
80C67⋊#95C2×C418yes—
81C67⋊#95D414yes—
82C23×C4×C67C2312yes—
83C23×C4×C67C2312yes—
84C23×C4×C67C2311yescharacteristic
85C23×C4×C67C2311yescharacteristic
86C23×C4×C67C2311yescharacteristic
87C2×C42×C67—22no—
88C2×C42×C67—22no—
89C2×C42×C67C2×C414yes—
90#95×C67—24no—
91#95×C67—22no—
92#95×C67—22no—
93#95×C67C2×C418yes—
94C22×Dic67—22no—
95C22×Dic67—21no—
96C22×Dic67—21no—
97C22×Dic67—22no—
98C22×Dic67—21no—
99C22×Dic67—21no—
100C22×Dic67—22no—
101C22×Dic67—22no—
102C22×Dic67—21no—
103C22×Dic67—22no—
104C22×Dic67—21no—
105C22×Dic67—22no—
106C22×Dic67—22no—
107C22×Dic67—22no—
108C22×Dic67—22no—
109C22×Dic67—22no—
110C22×Dic67—22no—
111C22×Dic67—21no—
112C22×Dic67—22no—
113C22×Dic67—22no—
114C22×Dic67—21no—
115C22×Dic67C22×C412yes—
116C22×Dic67C22×C414yes—
117C22×Dic67C22×C412yes—
118C22×Dic67C22×C414yes—
119C22×Dic67C22×C412yes—
120C22×Dic67C22×C414yes—
121C22×Dic67C22×C412yes—
122C22×Dic67C22×C414yes—
123C22×Dic67C2×D412yes—
124C22×Dic67C2×D414yes—
125C22×Dic67C2×D412yes—
126C22×Dic67C2×Q812yes—
127C22×Dic67C2×Q812yes—
128C22×Dic67#4114yes—
129C22×Dic67#4114yes—
130C4⋊Dic67—216no—
131Dic67⋊C4—216no—
132Dic67⋊C4—216no—
133Dic67⋊C4C42116yescomplemented
134Dic67⋊C4C4⋊C4116yescomplemented
135C4×Dic67—28no—
136C4×Dic67—28no—
137C4×Dic67—216no—
138C4×Dic67—48no—
139C4×Dic67—48no—
140C24×C67C2411yescharacteristic
141C22×C4×C67—21no—
142C22×C4×C67—21no—
143C22×C4×C67—21no—
144C22×C4×C67—21no—
145C22×C4×C67—22no—
146C22×C4×C67—22no—
147C22×C4×C67—21no—
148C22×C4×C67—22no—
149C22×C4×C67—21no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—22no—
157C22×C4×C67—22no—
158C22×C4×C67—22no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67C22×C414yes—
163C22×C4×C67C22×C414yes—
164C22×C4×C67C22×C412yes—
165C22×C4×C67C22×C414yes—
166C22×C4×C67C22×C412yes—
167C22×C4×C67C22×C412yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×Q812yes—
171C22×C4×C67#4112yes—
172C42×C67—28no—
173C23×C67C23×C411yescharacteristic
174C23×C67C23×C412yes—
175C23×C67C22×D411yescharacteristic
176C23×C67C22×D411yescharacteristic
177C23×C67C22×Q811yescharacteristic
178C23×C67C2×#4112yes—
179C23×C67C2×#4112yes—
180C23×C67C2×#4111yescharacteristic
181C23×C67H(5,𝔽2)11yescharacteristic
182C23×C672−1+412yes—
183C23×C672−1+411yescharacteristic
184C2×C4×C67—22no—
185C2×C4×C67—22no—
186C2×C4×C67—22no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—22no—
190C2×C4×C67—22no—
191C2×C4×C67—22no—
192C2×C4×C67—22no—
193C2×C4×C67—24no—
194C2×C4×C67—22no—
195C2×C4×C67—22no—
196C2×C4×C67—24no—
197C2×C4×C67—22no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—24no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—42no—
205C2×C4×C67—42no—
206C2×C4×C67—42no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67C2×C4214yes—
213C2×C4×C67C2×C4⋊C414yes—
214C2×C4×C67C4×D414yes—
215C2×Dic67—24no—
216C2×Dic67—22no—
217C2×Dic67—22no—
218C2×Dic67—24no—
219C2×Dic67—24no—
220C2×Dic67—22no—
221C2×Dic67—22no—
222C2×Dic67—24no—
223C2×Dic67—22no—
224C2×Dic67—24no—
225C2×Dic67—24no—
226C2×Dic67—22no—
227C2×Dic67—24no—
228C2×Dic67—22no—
229C2×Dic67—24no—
230C2×Dic67—22no—
231C2×Dic67—22no—
232C2×Dic67—24no—
233C2×Dic67—24no—
234C2×Dic67—22no—
235C2×Dic67—22no—
236C2×Dic67—24no—
237C2×Dic67—42no—
238C2×Dic67—42no—
239C2×Dic67—42no—
240C2×Dic67—42no—
241C2×Dic67—42no—
242C2×Dic67—42no—
243C2×Dic67C2×C4214yes—
244C2×Dic67C2×C4214yes—
245C2×Dic67C2×C4⋊C414yes—
246C2×Dic67C2×C4⋊C414yes—
247C2×Dic67#11714yes—
248C2×Dic67C4×D418yes—
249C2×Dic67C4×D418yes—
250C2×Dic67C4×Q818yes—
251C22×C67C22×C4211yescharacteristic
252C22×C67C22×C4⋊C411yescharacteristic
253C22×C67C2×#11711yescharacteristic
254C22×C67C2×C4×D412yes—
255C22×C67C2×C4×D412yes—
256C22×C67C2×C4×D411yescharacteristic
257C22×C67C2×C4×Q812yes—
258C22×C67C4×#4112yes—
259C22×C67C4×#4111yescharacteristic
260C22×C67#51911yescharacteristic
261C22×C67#51912yes—
262C22×C67#52011yescharacteristic
263C22×C67#52012yes—
264C22×C67D4211yescharacteristic
265C22×C67#54712yes—
266C22×C67#54711yescharacteristic
267C22×C67#54812yes—
268C22×C67D4×Q811yescharacteristic
269C22×C67#55212yes—
270C22×C67#55312yes—
271C22×C67#55411yescharacteristic
272C22×C67#55612yes—
273C22×C67#55712yes—
274C4×C67—24no—
275C4×C67—24no—
276C4×C67—24no—
277C4×C67—44no—
278C4×C67—44no—
279Dic67—24no—
280Dic67—28no—
281Dic67—24no—
282Dic67—42no—
283Dic67—44no—
284Dic67C4×C4⋊C418yescomplemented
285#13821—671nomaximal, Sylow
286C2×C67C2×C4×C4⋊C411yescharacteristic
287C2×C67C42×D411yescharacteristic
288C2×C67#227611yescharacteristic
289C2×C67#234411yescharacteristic
290C2×C67#234711yescharacteristic
291C2×C67D4×C4⋊C411yescharacteristic
292C2×C67#235111yescharacteristic
293C2×C67#236112yes—
294C2×C67#236212yes—
295C2×C67#236512yes—
296C2×C67#237012yes—
297C2×#389—672no—
298C2×C4×C4⋊C4—671no—
299C2×C4×C4⋊C4—671no—
300C2×C4×C4⋊C4—672no—
301C2×C4×C4⋊C4—671no—
302#1436—672no—
303#1436—672no—
304C4⋊#95—671no—
305#1434—671no—
306#1434D6711yescharacteristic
307C4×#95—671no—
308C67#1382111yescharacteristic, complemented, Sylow
309C22×C4⋊C4—671no—
310C22×C4⋊C4—671no—
311C22×C4⋊C4—672no—
312C22×C4⋊C4—672no—
313C22×C4⋊C4—672no—
314C22×C42—672no—
315C22×C42—672no—
316C22×C42—671no—
317C22×C42—672no—
318C22×C42—671no—
319C22×C42C2×D6711yescharacteristic
320#389—6716no—
321C4×C4⋊C4—678no—
322C4×C4⋊C4—1344no—
323C4×C4⋊C4—1348no—
324C4×C4⋊C4—1344no—
325C2×#95—672no—
326C2×#95—671no—
327C2×#95—672no—
328C2×#95—671no—
329C2×#95—671no—
330C2×#95—672no—
331C2×#95—671no—
332C2×#95—672no—
333C2×#95C2×D6711yescharacteristic
334C2×#95C2×D6711yescharacteristic
335C2×#95C2×D6712yes—
336C2×#95C2×D6712yes—
337C23×C4—671no—
338C23×C4—671no—
339C23×C4—672no—
340C23×C4—671no—
341C23×C4—672no—
342C23×C4—671no—
343C23×C4C22×D6712yes—
344C23×C4C22×D6712yes—
345C23×C4C22×D6711yescharacteristic
346C23×C4C22×D6711yescharacteristic
347C23×C4C22×D6711yescharacteristic
348C2×C4⋊C4—674no—
349C2×C4⋊C4—674no—
350C2×C4⋊C4—678no—
351C2×C4⋊C4—674no—
352C2×C4⋊C4—674no—
353C2×C4⋊C4—678no—
354C2×C4⋊C4—678no—
355C2×C4⋊C4—678no—
356C2×C4⋊C4—1344no—
357C2×C4⋊C4—1344no—
358C2×C4⋊C4—1344no—
359C2×C4⋊C4—1344no—
360C2×C4⋊C4—1344no—
361C2×C4⋊C4—1344no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—2682no—
365C2×C42—2682no—
366C2×C42—674no—
367C2×C42—674no—
368C2×C42—678no—
369C2×C42—1344no—
370C2×C42—1344no—
371C2×C42—1344no—
372C2×C42—1344no—
373C2×C42—1344no—
374C2×C42—1344no—
375C2×C42—1344no—
376C2×C42—1342no—
377C2×C42—1342no—
378C2×C42C4×D6714yes—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—2682no—
383#95—674no—
384#95—674no—
385#95—678no—
386#95—674no—
387#95—1344no—
388#95—1344no—
389#95—1342no—
390#95C4×D6718yes—
391C24C23×D6711yescharacteristic
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—672no—
414C22×C4—674no—
415C22×C4—672no—
416C22×C4—672no—
417C22×C4—674no—
418C22×C4—672no—
419C22×C4—672no—
420C22×C4—674no—
421C22×C4—674no—
422C22×C4—674no—
423C22×C4—672no—
424C22×C4—672no—
425C22×C4—674no—
426C22×C4—672no—
427C22×C4—674no—
428C22×C4—1342no—
429C22×C4—1342no—
430C22×C4—1342no—
431C22×C4—1342no—
432C22×C4—1342no—
433C22×C4—1342no—
434C22×C4—1341no—
435C22×C4—1342no—
436C22×C4—1342no—
437C22×C4—1341no—
438C22×C4—1342no—
439C22×C4—1341no—
440C22×C4—1341no—
441C22×C4—1342no—
442C22×C4—1342no—
443C22×C4—1341no—
444C22×C4—1341no—
445C22×C4—1342no—
446C22×C4—1341no—
447C22×C4—1342no—
448C22×C4—1341no—
449C22×C4C2×C4×D6714yes—
450C22×C4C2×C4×D6714yes—
451C22×C4C2×C4×D6712yes—
452C22×C4C2×C4×D6714yes—
453C22×C4C2×C4×D6712yes—
454C22×C4C2×C4×D6712yes—
455C22×C4D4×D6712yes—
456C22×C4D4×D6712yes—
457C22×C4Q8×D6712yes—
458C22×C4#1176162712yes—
459C4⋊C4—6716no—
460C4⋊C4—6716no—
461C4⋊C4—13416no—
462C4⋊C4—13416no—
463C4⋊C4—13416no—
464C42—28no—
465C42—2688no—
466C42—2688no—
467C42—1348no—
468C42—13416no—
469C42—1348no—
470C23C22×C4×D6711yescharacteristic
471C23C22×C4×D6712yes—
472C23C2×D4×D6711yescharacteristic
473C23C2×D4×D6711yescharacteristic
474C23C2×Q8×D6711yescharacteristic
475C23#41×D6712yes—
476C23#41×D6712yes—
477C23C2×#1176162711yescharacteristic
478C23#1452408911yescharacteristic
479C23#1452409012yes—
480C23#1452409011yescharacteristic
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—2682no—
510C2×C4—2682no—
511C2×C4—2682no—
512C2×C4—2682no—
513C2×C4—2682no—
514C2×C4—2682no—
515C2×C4—674no—
516C2×C4—674no—
517C2×C4—674no—
518C2×C4—678no—
519C2×C4—678no—
520C2×C4—674no—
521C2×C4—674no—
522C2×C4—678no—
523C2×C4—1342no—
524C2×C4—1344no—
525C2×C4—1344no—
526C2×C4—1342no—
527C2×C4—1342no—
528C2×C4—1344no—
529C2×C4—1342no—
530C2×C4—1344no—
531C2×C4—1344no—
532C2×C4—1342no—
533C2×C4—1344no—
534C2×C4—1342no—
535C2×C4—1344no—
536C2×C4—1342no—
537C2×C4—1344no—
538C2×C4—1344no—
539C2×C4—1342no—
540C2×C4—1342no—
541C2×C4—1344no—
542C2×C4—1342no—
543C2×C4—1342no—
544C2×C4—1344no—
545C2×C4C42×D6714yes—
546C2×C4C4⋊C4×D6714yes—
547C2×C4(C22⋊D67)⋊C414yes—
548C22C2×C42×D6711yescharacteristic
549C22C2×C4⋊C4×D6711yescharacteristic
550C22#117×D6711yescharacteristic
551C22C4×D4×D6712yes—
552C22C4×D4×D6712yes—
553C22C2×(C22⋊D67)⋊C411yescharacteristic
554C22C4×Q8×D6712yes—
555C22C4×#1176162712yes—
556C22#1686386411yescharacteristic
557C22#1686387311yescharacteristic
558C22#1686387312yes—
559C22#1686387611yescharacteristic
560C22#1686387612yes—
561C22#1686403011yescharacteristic
562C22#1686404412yes—
563C22#1686404611yescharacteristic
564C22#1686405412yes—
565C22#1686405911yescharacteristic
566C22#1686408312yes—
567C22#1686409712yes—
568C22#1686410411yescharacteristic
569C22#1686411312yes—
570C22#1686412312yes—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—44no—
575C4—44no—
576C4—2682no—
577C4—2684no—
578C4—678no—
579C4—1344no—
580C4—1348no—
581C4—1344no—
582C2C4×C4⋊C4×D6711yescharacteristic
583C2C4×(C22⋊D67)⋊C411yescharacteristic
584C2#2256257211yescharacteristic
585C2#2256307311yescharacteristic
586C2#2256309711yescharacteristic
587C2#2256310511yescharacteristic
588C2#2256312611yescharacteristic
589C2#2256322212yes—
590C2#2256323312yes—
591C2#2256326412yes—
592C2#2256330212yes—
subgroupG/H
center, FrattiniC24C23×D67
commutatorC22×C67C22×C42
FittingC67×#1434C2
socle, prime radical, cosocleC24×C67C24
hypercenter#1434D67
hypocenterC67#13821
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
22. #16863293C4semidirect
23. C4×Dic67⋊C4C4semidirect
133. Dic67⋊C4C42semidirect
134. Dic67⋊C4C4⋊C4semidirect
284. Dic67C4×C4⋊C4semidirect
308. C67#13821semidirect

Maximal subgroups

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

subgroupindex
C2×#168632932
C2×C4×Dic67⋊C42
C2×C4×C4⋊Dic672
C2×#168632472
#225584432
#225584402
Dic67⋊#952
#225584332
#95×Dic672
C67×#14342
#1382167