Finite Group Databasebeta

← #30821651

Subgroups — #30821651

Order 17152.

1278 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863877C211yesmaximal, characteristic
2#22569231C212yesmaximal, complemented
3#22558531C212yesmaximal, complemented
4#22558526C212yesmaximal, complemented
5#22558525C212yesmaximal, complemented
6#22558522C212yesmaximal, complemented
7C2×#16863122C211yesmaximal, characteristic, complemented
8#324×D67C212yesmaximal, complemented
9C2×#324×C67C211yesmaximal, characteristic, complemented
10C22×#11761628C411yescharacteristic
11#16863877—24no—
12#16863877—24no—
13C2×#14524008C411yescharacteristic
14C2×#14524008C2211yescharacteristic
15C2×#14524005C411yescharacteristic
16C2×#14524005C2211yescharacteristic
17C2×C4×C22⋊D67C411yescharacteristic
18C2×C4×C22⋊D67C411yescharacteristic
19C2×C4×C22⋊D67C2211yescharacteristic
20C2×C4×C22⋊D67C2211yescharacteristic
21C2×#14524002C411yescharacteristic
22C2×#14524001C411yescharacteristic
23C2×#14524001C2211yescharacteristic
24C2×C4⋊C4×D67C411yescharacteristic
25C2×C4⋊C4×D67C2211yescharacteristic
26C8×C22⋊D67—24no—
27(C22⋊D67)⋊C8—24no—
28#16863446C2214yes—
29C2×C22⋊(C67⋊C8)C2212yes—
30C22⋊C8×D67C2214yes—
31#16863122C2212yes—
32#16863122C2212yes—
33#16863121C2214yes—
34#16863119C2214yes—
35C22×C4⋊C4×C67C2211yescharacteristic
36C2×C22⋊C8×C67C2212yes—
37#324×C67C2212yes—
38#324×C67C2212yes—
39C2×#11761628—22no—
40C2×#11761628—22no—
41C2×#11761628—24no—
42C2×#11761628C818yescomplemented
43C22×Dic134C2×C411yescharacteristic
44C22×D268C2×C411yescharacteristic
45C22×C22⋊D67C2×C411yescharacteristic
46C22×C22⋊D67C2×C411yescharacteristic
47C22×C4×D67C2×C411yescharacteristic
48C22×C4×D67C2×C411yescharacteristic
49C22×C4×D67C2×C411yescharacteristic
50C22×C4×D67C2×C411yescharacteristic
51C22×C4×D67C2×C411yescharacteristic
52C22×C4×D67C2311yescharacteristic
53C22×C67⋊C8—22no—
54C2×C8×D67—24no—
55#14524008—24no—
56#14524008—24no—
57#14524005—24no—
58#14524005—24no—
59C4×C22⋊D67—22no—
60C4×C22⋊D67—22no—
61C4×C22⋊D67—22no—
62C4×C22⋊D67—22no—
63C4×C22⋊D67—22no—
64C4×C22⋊D67—22no—
65C4×C22⋊D67D414yes—
66C4×C22⋊D67D414yes—
67#14524002—22no—
68#14524002—22no—
69#14524001—24no—
70#14524001—24no—
71C2×C4⋊Dic67C2×C411yescharacteristic
72C2×C4⋊Dic67C2×C411yescharacteristic
73C2×Dic67⋊C4C2×C411yescharacteristic
74C2×Dic67⋊C4C2×C411yescharacteristic
75C2×Dic67⋊C4C2×C411yescharacteristic
76C2×Dic67⋊C4C2311yescharacteristic
77C4⋊C4×D67—24no—
78C4⋊C4×D67—24no—
79C2×C22⋊Dic67C2×C411yescharacteristic
80C2×C22⋊Dic67C2×C411yescharacteristic
81C2×(C2×D67)⋊C4C2×C411yescharacteristic
82C2×(C2×D67)⋊C4C2×C411yescharacteristic
83C2×(C2×D67)⋊C4C2×C411yescharacteristic
84C2×(C2×D67)⋊C4C2311yescharacteristic
85C2×C4×Dic67C2×C411yescharacteristic
86C2×C4×Dic67C2×C411yescharacteristic
87C2×C4×Dic67C2×C411yescharacteristic
88C2×C4×Dic67C2×C411yescharacteristic
89C2×C4×Dic67C2×C411yescharacteristic
90C2×C4×Dic67C2311yescharacteristic
91Dic67⋊C8—24no—
92C22⋊(C67⋊C8)—22no—
93C22⋊(C67⋊C8)C2314yes—
94(C2×D67)⋊C8—24no—
95C8×Dic67—24no—
96C23×C4×C67C2×C411yescharacteristic
97C23×C4×C67C2×C411yescharacteristic
98C23×C4×C67C2311yescharacteristic
99C22×C8×C67—22no—
100C2×C4⋊C4×C67—22no—
101C2×C4⋊C4×C67—22no—
102C2×C4⋊C4×C67C2×C411yescharacteristic
103C2×C4⋊C4×C67C2×C411yescharacteristic
104C2×C4⋊C4×C67C2311yescharacteristic
105C2×C4⋊C4×C67C2311yescharacteristic
106C22⋊C8×C67—22no—
107C22⋊C8×C67C2314yes—
108C23×D67C22×C411yescharacteristic
109C23×D67C22×C411yescharacteristic
110#11761628—216no—
111#11761628—216no—
112C2×Dic134—22no—
113C2×Dic134—22no—
114C2×Dic134C2×C814yescomplemented
115C2×D268—22no—
116C2×D268—22no—
117C2×D268C2×C814yescomplemented
118C2×C22⋊D67—21no—
119C2×C22⋊D67—21no—
120C2×C22⋊D67—22no—
121C2×C22⋊D67C22⋊C412yes—
122C2×C22⋊D67C22⋊C412yes—
123C2×C22⋊D67C2×C814yes—
124C2×C22⋊D67C2×C814yescomplemented
125C2×C22⋊D67M1614yes—
126C22×Dic67C22×C411yescharacteristic
127C22×Dic67C22×C411yescharacteristic
128C22×Dic67C22×C411yescharacteristic
129C22×Dic67C22×C411yescharacteristic
130C22×Dic67C22×C411yescharacteristic
131C22×Dic67C22×C411yescharacteristic
132C2×C4×D67—22no—
133C2×C4×D67—22no—
134C2×C4×D67—22no—
135C2×C4×D67—22no—
136C2×C4×D67—22no—
137C2×C4×D67—22no—
138C2×C4×D67—22no—
139C2×C4×D67—81no—
140C2×C4×D67—81no—
141C2×C4×D67—81no—
142C2×C4×D67—81no—
143C2×C4×D67C2×C814yescomplemented
144C2×C4×D67C2×C814yescomplemented
145C2×C4×D67C2×D414yes—
146C2×C67⋊C8—24no—
147C2×C67⋊C8—82no—
148C8×D67—84no—
149C4⋊Dic67—22no—
150C4⋊Dic67—22no—
151Dic67⋊C4—22no—
152Dic67⋊C4—22no—
153Dic67⋊C4—22no—
154Dic67⋊C4C22⋊C414yes—
155C22⋊Dic67—21no—
156C22⋊Dic67—21no—
157C22⋊Dic67C22⋊C412yes—
158C22⋊Dic67C22⋊C412yes—
159(C2×D67)⋊C4—22no—
160(C2×D67)⋊C4—22no—
161(C2×D67)⋊C4—22no—
162(C2×D67)⋊C4C22⋊C414yes—
163C4×Dic67—22no—
164C4×Dic67—22no—
165C4×Dic67—22no—
166C4×Dic67—22no—
167C4×Dic67—22no—
168C4×Dic67C2×D414yes—
169C24×C67C22×C411yescharacteristic
170C22×C4×C67—21no—
171C22×C4×C67—21no—
172C22×C4×C67—22no—
173C22×C4×C67—21no—
174C22×C4×C67—21no—
175C22×C4×C67—81no—
176C22×C4×C67—81no—
177C22×C4×C67C2×C814yescomplemented
178C22×C4×C67C22×C411yescharacteristic
179C22×C4×C67C22×C411yescharacteristic
180C22×C4×C67C22×C411yescharacteristic
181C22×C4×C67C22×C411yescharacteristic
182C22×C4×C67C22×C411yescharacteristic
183C22×C4×C67C2×D412yes—
184C22×C4×C67C2×D412yes—
185C22×C4×C67C2411yescharacteristic
186C2×C8×C67—24no—
187C2×C8×C67—82no—
188C4⋊C4×C67—22no—
189C4⋊C4×C67—22no—
190C4⋊C4×C67—22no—
191C4⋊C4×C67—22no—
192C23×C67—41no—
193C23×C67—41no—
194C23×C67C2×C22⋊C411yescharacteristic
195C23×C67C2×C22⋊C411yescharacteristic
196C23×C67C22×C812yes—
197C23×C67C2×M1612yes—
198C23×C67C23×C411yescharacteristic
199C22×D67—21no—
200C22×D67—21no—
201C22×D67—41no—
202C22×D67—41no—
203C22×D67—41no—
204C22×D67—41no—
205C22×D67C2×C22⋊C412yes—
206C22×D67C22×C812yes—
207C22×D67C22×C812yes—
208C22×D67C2×M1612yes—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—24no—
214C2×C4×C67—24no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—42no—
218C2×C4×C67—41no—
219C2×C4×C67—42no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—81no—
224C2×C4×C67—81no—
225C2×C4×C67C2×C22⋊C412yes—
226C2×C4×C67C22×C812yes—
227C2×C4×C67C22×C812yes—
228C2×C4×C67C22×D412yes—
229C22⋊D67—24no—
230C22⋊D67—24no—
231C22⋊D67C22⋊C818yes—
232C22⋊D67C22⋊C818yes—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236C2×Dic67—21no—
237C2×Dic67—41no—
238C2×Dic67—41no—
239C2×Dic67—41no—
240C2×Dic67—41no—
241C2×Dic67C2×C22⋊C412yes—
242C2×Dic67C2×C22⋊C412yes—
243C2×Dic67C2×C22⋊C412yes—
244C2×Dic67C22×C812yes—
245C2×Dic67C22×C812yes—
246C2×Dic67C2×M1612yes—
247D268—28no—
248C4×D67—28no—
249C4×D67—28no—
250C4×D67—44no—
251C4×D67—82no—
252C4×D67—82no—
253C4×D67—82no—
254Dic134—28no—
255C8×C67—44no—
256C67⋊C8—44no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67—21no—
260C22×C67—21no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—42no—
264C22×C67—42no—
265C22×C67C2×C22⋊C812yes—
266C22×C67C2×C22⋊C812yes—
267C22×C67C22×C22⋊C411yescharacteristic
268C22×C67C23×C811yescharacteristic
269C22×C67C22×M1611yescharacteristic
270C2×D67—22no—
271C2×D67—22no—
272C2×D67—22no—
273C2×D67—42no—
274C2×D67—42no—
275C2×D67—42no—
276C2×D67—41no—
277C2×D67—41no—
278C2×D67—42no—
279C2×D67C2×C22⋊C814yes—
280C4×C67—22no—
281C4×C67—22no—
282C4×C67—22no—
283C4×C67—22no—
284C4×C67—41no—
285C4×C67—41no—
286Dic67—22no—
287Dic67—22no—
288Dic67—22no—
289Dic67—41no—
290Dic67—41no—
291Dic67C2×C22⋊C814yes—
292#17351—671nomaximal, Sylow
293C2×C67—21no—
294C2×C67—21no—
295C2×C67—22no—
296C2×C67—22no—
297C2×C67C22×C22⋊C811yescharacteristic
298C2×C67#288311yescharacteristic
299C2×C67#288911yescharacteristic
300D67—24no—
301D67—42no—
302C2×#520—671no—
303#2972—672no—
304#1461—672no—
305#1459—672no—
306#1459—672no—
307#1458—672no—
308C2×#324—672no—
309C2×#324—671no—
310C2×#324D6711yescharacteristic, complemented
311C67#1735111yescharacteristic, complemented, Sylow
312C22×#41—671no—
313#520—1344no—
314#520—1344no—
315C2×C4×Q8—671no—
316C2×C4×Q8—671no—
317C2×C4×D4—671no—
318C2×C4×D4—671no—
319C2×C4×D4—671no—
320C2×C4×D4—671no—
321C2×C4×D4—671no—
322C2×C4×D4—671no—
323C2×#117—671no—
324C2×#117—671no—
325C2×#117—671no—
326C22×C4⋊C4—671no—
327C22×C4⋊C4—671no—
328C22×C4⋊C4C2×D6711yescharacteristic
329C8×D4—1344no—
330C8×D4—1344no—
331#431—674no—
332C2×C22⋊C8—672no—
333C2×C22⋊C8—674no—
334C2×C22⋊C8C2×D6712yes—
335#324—672no—
336#324—672no—
337#324—674no—
338#324C2×D6712yes—
339#324C2×D6712yes—
340#323—674no—
341C2×#41—678no—
342C2×#41—1342no—
343C2×#41—1342no—
344C2×#41—1344no—
345C22×Q8—671no—
346C22×D4—671no—
347C22×D4—671no—
348C22×D4—671no—
349C23×C4—671no—
350C23×C4—671no—
351C23×C4—671no—
352C23×C4—671no—
353C23×C4—671no—
354C23×C4—671no—
355C23×C4C4×D6711yescharacteristic
356C23×C4C4×D6711yescharacteristic
357C23×C4C22×D6711yescharacteristic
358C22×C8—22no—
359C22×C8—1344no—
360C22×C8—1342no—
361C4×Q8—1344no—
362C4×Q8—1344no—
363C4×D4—674no—
364C4×D4—674no—
365C4×D4—1344no—
366C4×D4—1342no—
367C4×D4—1342no—
368C4×D4—1344no—
369C4×D4—1342no—
370C4×D4—1342no—
371C4×D4—1342no—
372C4×D4—1342no—
373#117—1344no—
374#117—1344no—
375#117—1342no—
376#117—1342no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—671no—
380C2×C4⋊C4—671no—
381C2×C4⋊C4—671no—
382C2×C4⋊C4—671no—
383C2×C4⋊C4—671no—
384C2×C4⋊C4—671no—
385C2×C4⋊C4—1344no—
386C2×C4⋊C4—1344no—
387C2×C4⋊C4C4×D6711yescharacteristic
388C2×C4⋊C4C4×D6711yescharacteristic
389C2×C4⋊C4C22×D6711yescharacteristic
390C2×C4⋊C4C22×D6711yescharacteristic
391C2×C22⋊C4—671no—
392C2×C22⋊C4—671no—
393C2×C22⋊C4—671no—
394C2×C22⋊C4—671no—
395C2×C22⋊C4—671no—
396C2×C22⋊C4—671no—
397C2×C42—671no—
398C2×C42—671no—
399C2×C42—671no—
400C2×C42—671no—
401C2×C42—671no—
402C2×C42—671no—
403C4⋊C8—1344no—
404C22⋊C8—22no—
405C22⋊C8—674no—
406C22⋊C8—1344no—
407C22⋊C8—1342no—
408C22⋊C8C22×D6714yes—
409C4×C8—1344no—
410C24—671no—
411C24—671no—
412C24C2×C4×D6711yescharacteristic
413#41—13416no—
414#41—13416no—
415C2×Q8—674no—
416C2×Q8—1342no—
417C2×Q8—1342no—
418C2×D4—672no—
419C2×D4—672no—
420C2×D4—674no—
421C2×D4—674no—
422C2×D4—674no—
423C2×D4—674no—
424C2×D4—1342no—
425C2×D4—1342no—
426C2×D4—1341no—
427C2×D4—1341no—
428C2×D4—1342no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—5361no—
437C22×C4—5361no—
438C22×C4—5361no—
439C22×C4—5361no—
440C22×C4—671no—
441C22×C4—671no—
442C22×C4—674no—
443C22×C4—671no—
444C22×C4—671no—
445C22×C4—671no—
446C22×C4—671no—
447C22×C4—674no—
448C22×C4—674no—
449C22×C4—1342no—
450C22×C4—1342no—
451C22×C4—1342no—
452C22×C4—1342no—
453C22×C4—1342no—
454C22×C4—1342no—
455C22×C4—1342no—
456C22×C4C8×D6714yescomplemented
457C22×C4C2×C4×D6711yescharacteristic
458C22×C4C2×C4×D6711yescharacteristic
459C22×C4C2×C4×D6711yescharacteristic
460C22×C4C2×C4×D6711yescharacteristic
461C22×C4C2×C4×D6711yescharacteristic
462C22×C4D4×D6712yes—
463C22×C4D4×D6712yes—
464C22×C4C23×D6711yescharacteristic
465C2×C8—24no—
466C2×C8—82no—
467C2×C8—5364no—
468C2×C8—5362no—
469C2×C8—1344no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—674no—
475C4⋊C4—1342no—
476C4⋊C4—1342no—
477C4⋊C4—1342no—
478C4⋊C4—1342no—
479C4⋊C4—1342no—
480C22⋊C4—672no—
481C22⋊C4—672no—
482C22⋊C4—674no—
483C22⋊C4—1342no—
484C22⋊C4—1342no—
485C22⋊C4—1341no—
486C22⋊C4—1341no—
487C22⋊C4—1342no—
488C42—674no—
489C42—1342no—
490C42—1342no—
491C42—1342no—
492C42—1342no—
493C42—1342no—
494C23—41no—
495C23—41no—
496C23—2681no—
497C23—2681no—
498C23—2681no—
499C23—2681no—
500C23—672no—
501C23—672no—
502C23—672no—
503C23—672no—
504C23—1341no—
505C23—1341no—
506C23C22⋊C4×D6711yescharacteristic
507C23C22⋊C4×D6711yescharacteristic
508C23C2×C8×D6712yes—
509C23M16×D6712yes—
510C23C22×C4×D6711yescharacteristic
511Q8—1348no—
512D4—678no—
513D4—678no—
514D4—1348no—
515D4—1344no—
516D4—1344no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—2684no—
534C2×C4—2681no—
535C2×C4—2681no—
536C2×C4—2681no—
537C2×C4—2681no—
538C2×C4—5362no—
539C2×C4—5362no—
540C2×C4—5362no—
541C2×C4—672no—
542C2×C4—672no—
543C2×C4—672no—
544C2×C4—672no—
545C2×C4—672no—
546C2×C4—672no—
547C2×C4—1348no—
548C2×C4—1348no—
549C2×C4—1341no—
550C2×C4—1341no—
551C2×C4—1341no—
552C2×C4—1341no—
553C2×C4C22⋊C4×D6712yes—
554C2×C4C2×C8×D6712yes—
555C2×C4C2×C8×D6712yes—
556C2×C4C2×D4×D6712yes—
557C8—44no—
558C8—2684no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—2682no—
568C22—2682no—
569C22—2682no—
570C22—2682no—
571C22—2681no—
572C22—2681no—
573C22—674no—
574C22—1342no—
575C22—1342no—
576C22—1342no—
577C22C22⋊C8×D6712yes—
578C22C22⋊C8×D6712yes—
579C22C2×C22⋊C4×D6711yescharacteristic
580C22C22×C8×D6711yescharacteristic
581C22C2×M16×D6711yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—2681no—
589C4—2681no—
590C4—674no—
591C4—1342no—
592C4—1342no—
593C4—1342no—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—2682no—
599C2—1344no—
600C2C2×C22⋊C8×D6711yescharacteristic
601C2#2256866711yescharacteristic
602C2#2256868911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D67
commutatorC22×C67C23×C8
FrattiniC22×C4C23×D67
FittingC2×#324×C67C2
socleC22×C67C22×C22⋊C4
prime radicalC22×#11761628C4
cosocleC22×C4×C67C24
hypercenterC2×#324D67
hypocenterC67#17351
LCSG ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #22569231C2semidirect
3. #22558531C2semidirect
4. #22558526C2semidirect
5. #22558525C2semidirect
6. #22558522C2semidirect
7. C2×#16863122C2semidirect
8. #324×D67C2semidirect
9. C2×#324×C67C2semidirect
42. C2×#11761628C8semidirect
114. C2×Dic134C2×C8semidirect
117. C2×D268C2×C8semidirect
124. C2×C22⋊D67C2×C8semidirect
143. C2×C4×D67C2×C8semidirect
144. C2×C4×D67C2×C8semidirect
177. C22×C4×C67C2×C8semidirect
310. C2×#324D67semidirect
311. C67#17351semidirect
456. C22×C4C8×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168638772
#225692312
#225585312
#225585262
#225585252
#225585222
C2×#168631222
#324×D672
C2×#324×C672
#1735167