Finite Group Databasebeta

← #30826972

Subgroups — #30826972

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570280C211yesmaximal, characteristic
2#22569156C211yesmaximal, characteristic, complemented
3#22559111C211yesmaximal, characteristic, complemented
4#22559097C211yesmaximal, characteristic, complemented
5#22559085C211yesmaximal, characteristic, complemented
6#22559080C211yesmaximal, characteristic, complemented
7#22559059C211yesmaximal, characteristic, complemented
8#22559050C211yesmaximal, characteristic, complemented
9#22559026C211yesmaximal, characteristic, complemented
10C8×#11761618C211yesmaximal, characteristic, complemented
11#22558995C211yesmaximal, characteristic, complemented
12#22558681C211yesmaximal, characteristic, complemented
13#22558596C211yesmaximal, characteristic
14#22558506C211yesmaximal, characteristic, complemented
15C67×#2971C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863737C411yescharacteristic
18#16863737C411yescharacteristic
19#16863681C411yescharacteristic
20#16863676C411yescharacteristic
21#16863606C411yescharacteristic
22#16863601C411yescharacteristic
23#16863558C411yescharacteristic
24#16863558C411yescharacteristic
25#16863526C2211yescharacteristic
26#16863526C2211yescharacteristic
27#16863524—21no—
28#16863524—21no—
29C8×Dic134C2211yescharacteristic
30C8×Dic134C2211yescharacteristic
31C4×#11761618C2211yescharacteristic
32C4×#11761618C2211yescharacteristic
33C4×#11761616C2211yescharacteristic
34C4×#11761616C2211yescharacteristic
35#16863449C2211yescharacteristic
36#16863449C2211yescharacteristic
37#16863424—21no—
38C2×C4⋊(C67⋊C8)C2211yescharacteristic
39#16863354—21no—
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46#16863126C2211yescharacteristic
47#16863126C2211yescharacteristic
48#16863126C2211yescharacteristic
49#16863126C2211yescharacteristic
50D4⋊(C67⋊C8)C2211yescharacteristic
51D4⋊(C67⋊C8)C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863117C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56C4×#41×C67C2211yescharacteristic
57C8×Q8×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59#436×C67—21no—
60#436×C67—21no—
61#436×C67C2211yescharacteristic
62#436×C67C2211yescharacteristic
63#432×C67—21no—
64#432×C67C2211yescharacteristic
65C4×M16×C67—21no—
66C4×M16×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C22×C67⋊C8—21no—
72#14524043C2×C411yescharacteristic
73#14524043C2×C411yescharacteristic
74#14524035—21no—
75#14524022—21no—
76#14524022C2×C411yescharacteristic
77#14524022C2×C411yescharacteristic
78C4×Dic134—21no—
79C4×Dic134C2311yescharacteristic
80C4×Dic134C2311yescharacteristic
81#14524002—21no—
82#14523976C2×C411yescharacteristic
83#14523976C2×C411yescharacteristic
84#14523974C2×C411yescharacteristic
85#14523974C2×C411yescharacteristic
86C4⋊(C67⋊C8)—21no—
87C4⋊(C67⋊C8)—21no—
88C4⋊(C67⋊C8)C2×C411yescharacteristic
89C4⋊(C67⋊C8)C2×C411yescharacteristic
90C4⋊(C67⋊C8)C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94Dic67⋊C8—21no—
95Dic67⋊C8—21no—
96#14523967C2×C411yescharacteristic
97#14523967C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×C411yescharacteristic
100#14523966C2×C411yescharacteristic
101#14523966C2×C411yescharacteristic
102#14523964C2×C411yescharacteristic
103#14523964C2×C411yescharacteristic
104C22⋊(C67⋊C8)—21no—
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106C22⋊(C67⋊C8)C2×C411yescharacteristic
107#14523953—21no—
108#14523953—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C2×#41×C67C2×C411yescharacteristic
116C2×M16×C67—21no—
117C2×M16×C67—21no—
118C2×M16×C67D411yescharacteristic
119C2×M16×C67D411yescharacteristic
120C4×Q8×C67C2×C411yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C4×D4×C67—21no—
123C4×D4×C67—21no—
124C4×D4×C67C2×C411yescharacteristic
125C4×D4×C67C2311yescharacteristic
126#117×C67—21no—
127#117×C67C2×C411yescharacteristic
128C2×C42×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137#97×C67—21no—
138#97×C67D411yescharacteristic
139#97×C67D411yescharacteristic
140C4×C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147#11761618—22no—
148#11761618—22no—
149#11761616—41no—
150#11761616—41no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8—42no—
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C4⋊Dic67—21no—
161C4⋊Dic67—21no—
162C4⋊Dic67—21no—
163C4⋊Dic67C22×C411yescharacteristic
164C4⋊Dic67C22×C411yescharacteristic
165Dic67⋊C4—21no—
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169C22⋊Dic67—21no—
170C22⋊Dic67—21no—
171C4×Dic67—21no—
172C4×Dic67—21no—
173#41×C67—22no—
174#41×C67—41no—
175C2×Q8×C67C22×C411yescharacteristic
176C2×D4×C67—21no—
177C2×D4×C67C22×C411yescharacteristic
178C22×C4×C67—21no—
179C22×C4×C67—21no—
180C22×C4×C67—21no—
181C22×C4×C67C22×C411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67#4111yescharacteristic
184M16×C67—22no—
185M16×C67—41no—
186M16×C67—42no—
187M16×C67—42no—
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67C2×D411yescharacteristic
191C2×C8×C67C2×D411yescharacteristic
192C2×C8×C67C2×D411yescharacteristic
193C2×C8×C67C2×D411yescharacteristic
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67C22×C411yescharacteristic
197C4⋊C4×C67C22×C411yescharacteristic
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67C22×C411yescharacteristic
205C42×C67C2×D411yescharacteristic
206C42×C67#4111yescharacteristic
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67C4×D411yescharacteristic
210Q8×C67—21no—
211Q8×C67#13112yes—
212D4×C67—21no—
213D4×C67—21no—
214D4×C67—42no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—41no—
223C2×C4×C67—42no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—81no—
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C4×D411yescharacteristic
229C2×C4×C67C4×D411yescharacteristic
230C2×C4×C67C23×C411yescharacteristic
231C2×C4×C67C22×D411yescharacteristic
232C2×C4×C67C2×#4111yescharacteristic
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236C2×Dic67—21no—
237Dic134—21no—
238Dic134—21no—
239Dic134—21no—
240Dic134—21no—
241Dic134—81no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—41no—
246C8×C67—41no—
247C8×C67#13712yes—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—21no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C67⋊C8—41no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67C2×C4×D411yescharacteristic
263C4×C67—21no—
264C4×C67—22no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—21no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67C2×#13711yescharacteristic
273C4×C67#57511yescharacteristic
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277Dic67—41no—
278#18013—671nomaximal, Sylow
279C2×C67—41no—
280C2×C67—81no—
281C2×C67#293311yescharacteristic
282C2×C67C4×#13711yescharacteristic
283C2×C67#295711yescharacteristic
284#3107—671no—
285#2971D6711yescharacteristic
286#2962—671no—
287#1592—671no—
288#1590—671no—
289#1588—671no—
290#1587—671no—
291#1584—671no—
292#1583—671no—
293#1579—671no—
294C8×Q16—671no—
295#1573—671no—
296#1497—671no—
297#1477—671no—
298#1453—671no—
299C67#1801311yescharacteristic, complemented, Sylow
300#533—671no—
301C4×#41C2×D6711yescharacteristic
302#485—671no—
303#485—671no—
304#475—671no—
305#474—671no—
306#462—671no—
307#461—671no—
308#452—671no—
309#452—671no—
310#446—671no—
311#446—671no—
312#446—1341no—
313#446—1341no—
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8C2×D6711yescharacteristic
317C8×Q8C2×D6711yescharacteristic
318C4×Q16—671no—
319C4×Q16—671no—
320C4×SD16—671no—
321C4×SD16—671no—
322#436—21no—
323#436—21no—
324#436C2×D6711yescharacteristic
325#436C2×D6711yescharacteristic
326#432—21no—
327#432C2×D6711yescharacteristic
328#431—671no—
329#431—671no—
330#424—1341no—
331C2×C4⋊C8—671no—
332#405—1341no—
333C4×M16—21no—
334C4×M16C2×D6711yescharacteristic
335#335—671no—
336#335—671no—
337#334—671no—
338#334—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344Q8⋊C8—671no—
345D4⋊C8—671no—
346D4⋊C8—671no—
347#322—671no—
348#322—671no—
349#322—671no—
350C82—671no—
351C2×#41C4×D6711yescharacteristic
352C2×Q16—671no—
353C2×Q16—671no—
354C2×SD16—671no—
355C2×SD16—671no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16C22⋊D6711yescharacteristic
359C2×M16C22⋊D6711yescharacteristic
360C22×C8—1341no—
361C4⋊Q8—671no—
362C4⋊Q8—671no—
363#125—1341no—
364C22⋊Q8—671no—
365C22⋊Q8—671no—
366C22⋊Q8—1341no—
367C4×Q8—671no—
368C4×Q8—671no—
369C4×Q8—1341no—
370C4×Q8C4×D6711yescharacteristic
371C4×Q8C22×D6711yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C4×D6711yescharacteristic
375C4×D4C22×D6711yescharacteristic
376#117—21no—
377#117—1341no—
378#117C4×D6711yescharacteristic
379C2×C42—21no—
380C2×C42C22×D6711yescharacteristic
381#107—671no—
382#107—671no—
383#106—671no—
384#106—671no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—671no—
390C4⋊C8—671no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8—1341no—
397C4⋊C8C22×D6711yescharacteristic
398C4⋊C8C22×D6711yescharacteristic
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#103—671no—
405#102—671no—
406#102—671no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—671no—
411C22⋊C8—671no—
412C22⋊C8—1341no—
413#97—21no—
414#97—1341no—
415#97—1341no—
416#97C22⋊D6711yescharacteristic
417#97C22⋊D6711yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—671no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8C22×D6711yescharacteristic
427C4×C8C22×D6711yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—671no—
431C2×Q8—671no—
432C2×Q8—1341no—
433C2×Q8C2×C4×D6711yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D6711yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C2×C4×D6711yescharacteristic
440C22×C4C2×C22⋊D6711yescharacteristic
441C22×C4#1176162811yescharacteristic
442Q16—1342no—
443Q16—1342no—
444SD16—2681no—
445SD16—2681no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—42no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—2682no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8—1341no—
460C2×C8—1341no—
461C2×C8C2×C22⋊D6711yescharacteristic
462C2×C8C2×C22⋊D6711yescharacteristic
463C2×C8C2×C22⋊D6711yescharacteristic
464C2×C8C2×C22⋊D6711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—671no—
468C4⋊C4—671no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C4⋊C4C2×C4×D6711yescharacteristic
477C4⋊C4C2×C4×D6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—1341no—
482C22⋊C4—1341no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—1341no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42C2×C22⋊D6711yescharacteristic
490C42#1176162811yescharacteristic
491C42C23×D6711yescharacteristic
492C23—21no—
493C23C4×C22⋊D6711yescharacteristic
494Q8—21no—
495Q8—5361no—
496Q8—1341no—
497Q8—1341no—
498Q8—1341no—
499Q8—1341no—
500Q8#1452405112yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4—1341no—
519C2×C4—1341no—
520C2×C4C4×C22⋊D6711yescharacteristic
521C2×C4C4×C22⋊D6711yescharacteristic
522C2×C4C4×C22⋊D6711yescharacteristic
523C2×C4C22×C4×D6711yescharacteristic
524C2×C4C22×C22⋊D6711yescharacteristic
525C2×C4C2×#1176162811yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—2681no—
532C8—2681no—
533C8—2681no—
534C8—2681no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C8—1341no—
539C8#1452407312yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×C4×C22⋊D6711yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—2681no—
555C4—2681no—
556C4—2681no—
557C4—2681no—
558C4C2×#1452405111yescharacteristic
559C4#1686416211yescharacteristic
560C4C2×#1452407311yescharacteristic
561C4#1686419411yescharacteristic
562C2—41no—
563C2—81no—
564C2#2256893411yescharacteristic
565C2C4×#1452407311yescharacteristic
566C2#2256912411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2971C2
socleC22×C67C2×C4×D4
prime radicalC2×#41×C67C2×C4
cosocleC42×C67C24
hypercenter#2971D67
hypocenterC67#18013
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #22569156C2semidirect
3. #22559111C2semidirect
4. #22559097C2semidirect
5. #22559085C2semidirect
6. #22559080C2semidirect
7. #22559059C2semidirect
8. #22559050C2semidirect
9. #22559026C2semidirect
10. C8×#11761618C2semidirect
11. #22558995C2semidirect
12. #22558681C2semidirect
14. #22558506C2semidirect
299. C67#18013semidirect

Maximal subgroups

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

subgroupindex
#225702802
#225691562
#225591112
#225590972
#225590852
#225590802
#225590592
#225590502
#225590262
C8×#117616182
#225589952
#225586812
#225585962
#225585062
C67×#29712
#1801367