Finite Group Databasebeta

← #30826953

Subgroups — #30826953

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#22570285C211yesmaximal, characteristic
2M16×Dic134C211yesmaximal, characteristic, complemented
3#22559116C211yesmaximal, characteristic, complemented
4#22559097C211yesmaximal, characteristic, complemented
5#22559078C211yesmaximal, characteristic, complemented
6#22559066C211yesmaximal, characteristic, complemented
7#22559059C211yesmaximal, characteristic, complemented
8#22559050C211yesmaximal, characteristic, complemented
9#22559021C211yesmaximal, characteristic, complemented
10C8×#11761618C211yesmaximal, characteristic, complemented
11#22558997C211yesmaximal, characteristic, complemented
12#22558674C211yesmaximal, characteristic
13#22558603C211yesmaximal, characteristic, complemented
14#22558506C211yesmaximal, characteristic, complemented
15C67×#2969C211yesmaximal, characteristic
16C2×C4×Dic134C2211yescharacteristic
17#16863724C411yescharacteristic
18#16863707C411yescharacteristic
19#16863700C411yescharacteristic
20#16863700C411yescharacteristic
21#16863615C411yescharacteristic
22#16863615C411yescharacteristic
23#16863551C411yescharacteristic
24#16863534C411yescharacteristic
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#16863420C2211yescharacteristic
38#16863419—21no—
39M16×Dic67—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
56#529×C67C2211yescharacteristic
57#446×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59#436×C67—21no—
60#436×C67C2211yescharacteristic
61#435×C67—21no—
62C8×D4×C67C2211yescharacteristic
63#433×C67—21no—
64#431×C67C2211yescharacteristic
65#405×C67—21no—
66C4×M16×C67C2211yescharacteristic
67C22×Dic134C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761618C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C2×#11761616C2×C411yescharacteristic
72C2×#11761612—21no—
73C4×Dic134—21no—
74C4×Dic134—21no—
75C4×Dic134C2×C411yescharacteristic
76C4×Dic134C2×C411yescharacteristic
77C4×Dic134C2311yescharacteristic
78C4×Dic134C2311yescharacteristic
79C2×C4⋊Dic67C2×C411yescharacteristic
80C2×Dic67⋊C4—21no—
81C2×C4×Dic67—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×M16×C67—21no—
116C2×M16×C67D411yescharacteristic
117C2×M16×C67D411yescharacteristic
118C22×C8×C67—21no—
119#126×C67—21no—
120#125×C67C2×C411yescharacteristic
121#124×C67C2×C411yescharacteristic
122#123×C67—21no—
123C22⋊Q8×C67C2×C411yescharacteristic
124#121×C67C2×C411yescharacteristic
125C4×Q8×C67C2311yescharacteristic
126C4×D4×C67—21no—
127C4×D4×C67C2311yescharacteristic
128#117×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—22no—
145C2×Dic134—21no—
146C2×Dic134—41no—
147C2×Dic134C22×C411yescharacteristic
148C2×Dic134C22×C411yescharacteristic
149C22×Dic67—21no—
150C22×Dic67—21no—
151#11761618—22no—
152#11761618—41no—
153#11761616—22no—
154#11761616—41no—
155#11761612—42no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C4⋊Dic67—21no—
165C4⋊Dic67C22×C411yescharacteristic
166C4⋊Dic67C22×C411yescharacteristic
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171C4×Dic67—21no—
172C4×Dic67—21no—
173C4×Dic67—21no—
174C4×Dic67—21no—
175C2×Q8×C67C22×C411yescharacteristic
176C2×D4×C67—21no—
177C2×D4×C67C22×C411yescharacteristic
178C22×C4×C67—21no—
179C22×C4×C67C22×C411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67#4111yescharacteristic
182M16×C67—22no—
183M16×C67—41no—
184M16×C67—42no—
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—42no—
188C2×C8×C67C2×D411yescharacteristic
189C2×C8×C67C2×D411yescharacteristic
190C2×C8×C67C2×D411yescharacteristic
191C2×C8×C67C2×D411yescharacteristic
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
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—
201C22⋊C4×C67—21no—
202C22⋊C4×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—21no—
212D4×C67—21no—
213D4×C67—21no—
214D4×C67—81no—
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—41no—
224C2×C4×C67—41no—
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C23×C411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—42no—
236C2×Dic67—81no—
237Dic134—21no—
238Dic134—21no—
239Dic134—21no—
240Dic134—42no—
241Dic134#13112yes—
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—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67C2×#13711yescharacteristic
273C4×C67#57511yescharacteristic
274Dic67—22no—
275Dic67—41no—
276Dic67—41no—
277Dic67—41no—
278#18012—671nomaximal, Sylow
279C2×C67—41no—
280C2×C67—81no—
281C2×C67#293311yescharacteristic
282C2×C67C4×#13711yescharacteristic
283C2×C67#295711yescharacteristic
284#3108—671no—
285#2969D6711yescharacteristic
286Q8×M16—671no—
287#1593—671no—
288#1590—671no—
289#1587—671no—
290#1585—671no—
291#1584—671no—
292#1583—671no—
293SD16⋊C8—671no—
294C8×Q16—671no—
295#1574—671no—
296#1495—671no—
297#1479—671no—
298#1453—671no—
299C67#1801211yescharacteristic, complemented, Sylow
300#529C2×D6711yescharacteristic
301C2×C4×Q8—671no—
302#483—671no—
303#480—671no—
304#479—671no—
305#479—671no—
306#464—671no—
307#464—671no—
308#451—671no—
309#448—671no—
310#446—671no—
311#446—671no—
312#446—1341no—
313#446—1341no—
314#446C2×D6711yescharacteristic
315C8×Q8—671no—
316C8×Q8—671no—
317C8×Q8C2×D6711yescharacteristic
318C4×Q16—671no—
319C4×Q16—671no—
320C4×SD16—671no—
321C4×SD16—671no—
322#436—21no—
323#436C2×D6711yescharacteristic
324#435—21no—
325C8×D4C2×D6711yescharacteristic
326#433—21no—
327#431—671no—
328#431—671no—
329#431C2×D6711yescharacteristic
330#423—671no—
331#423—1341no—
332#405—21no—
333C4×M16—1341no—
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—
351C22×Q8—671no—
352C2×Q16—671no—
353C2×Q16—671no—
354C2×SD16—671no—
355C2×SD16—671no—
356C2×M16—21no—
357C2×M16—1341no—
358C2×M16C22⋊D6711yescharacteristic
359C2×M16C22⋊D6711yescharacteristic
360C22×C8—21no—
361#126—21no—
362#125C4×D6711yescharacteristic
363#124C4×D6711yescharacteristic
364#123—21no—
365C22⋊Q8C4×D6711yescharacteristic
366#121C4×D6711yescharacteristic
367C4×Q8—671no—
368C4×Q8—671no—
369C4×Q8—671no—
370C4×Q8—671no—
371C4×Q8—1341no—
372C4×Q8—1341no—
373C4×Q8C22×D6711yescharacteristic
374C4×D4—21no—
375C4×D4C22×D6711yescharacteristic
376#117—21no—
377C2×C4⋊C4—671no—
378C2×C4⋊C4—1341no—
379C2×C42—1341no—
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
428C2×Q8—2681no—
429C2×Q8—671no—
430C2×Q8—671no—
431C2×Q8—1342no—
432C2×Q8—1341no—
433C2×Q8C2×C4×D6711yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D6711yescharacteristic
436C22×C4—21no—
437C22×C4—1341no—
438C22×C4—1341no—
439C22×C4C2×C4×D6711yescharacteristic
440C22×C4C2×C22⋊D6711yescharacteristic
441C22×C4#1176162811yescharacteristic
442Q16—2681no—
443Q16—1342no—
444SD16—2681no—
445SD16—1342no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—2682no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
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—21no—
468C4⋊C4—21no—
469C4⋊C4—671no—
470C4⋊C4—671no—
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—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—1341no—
485C42—1341no—
486C42—1341no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42C2×C22⋊D6711yescharacteristic
490C42#1176162811yescharacteristic
491C42C23×D6711yescharacteristic
492C23—21no—
493C23C4×C22⋊D6711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—2682no—
497Q8—672no—
498Q8—1341no—
499Q8—1341no—
500Q8—1341no—
501D4—21no—
502D4—21no—
503D4—81no—
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—41no—
513C2×C4—41no—
514C2×C4—2682no—
515C2×C4—5361no—
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—41no—
552C4—41no—
553C4—41no—
554C4—2681no—
555C4—2681no—
556C4—2681no—
557C4—1342no—
558C4#131×D6711yescharacteristic
559C4#1686416111yescharacteristic
560C4C2×#1452407311yescharacteristic
561C4#1686419411yescharacteristic
562C2—41no—
563C2—81no—
564C2#2256893111yescharacteristic
565C2C4×#1452407311yescharacteristic
566C2#2256912411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2969C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#2969D67
hypocenterC67#18012
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×Dic134C2semidirect
3. #22559116C2semidirect
4. #22559097C2semidirect
5. #22559078C2semidirect
6. #22559066C2semidirect
7. #22559059C2semidirect
8. #22559050C2semidirect
9. #22559021C2semidirect
10. C8×#11761618C2semidirect
11. #22558997C2semidirect
13. #22558603C2semidirect
14. #22558506C2semidirect
299. C67#18012semidirect

Maximal subgroups

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

subgroupindex
#225702852
M16×Dic1342
#225591162
#225590972
#225590782
#225590662
#225590592
#225590502
#225590212
C8×#117616182
#225589972
#225586742
#225586032
#225585062
C67×#29692
#1801267