Finite Group Databasebeta

← #30827225

Subgroups — #30827225

Order 17152.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570248C211yesmaximal, characteristic
2#22569310C211yesmaximal, characteristic
3#22559116C211yesmaximal, characteristic, complemented
4#22559104C211yesmaximal, characteristic, complemented
5#22559080C211yesmaximal, characteristic, complemented
6#22559078C211yesmaximal, characteristic, complemented
7#22559059C211yesmaximal, characteristic, complemented
8#22559057C211yesmaximal, characteristic, complemented
9#22559026C211yesmaximal, characteristic, complemented
10C8×#11761618C211yesmaximal, characteristic, complemented
11#22558990C211yesmaximal, characteristic
12#22558681C211yesmaximal, characteristic
13#22558604C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic
15C67×#2984C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863731C411yescharacteristic
18#16863731C411yescharacteristic
19#16863691C411yescharacteristic
20#16863691C411yescharacteristic
21#16863606C411yescharacteristic
22#16863606C411yescharacteristic
23#16863552C411yescharacteristic
24#16863552C411yescharacteristic
25#16863526C2211yescharacteristic
26#16863526C2211yescharacteristic
27#16863523—21no—
28C8×Dic134C2211yescharacteristic
29C8×Dic134C2211yescharacteristic
30#16863520—21no—
31C4×#11761618C2211yescharacteristic
32C4×#11761618C2211yescharacteristic
33C4×#11761618C2211yescharacteristic
34C4×#11761618C2211yescharacteristic
35#16863454—21no—
36#16863454—21no—
37#16863449C2211yescharacteristic
38#16863449C2211yescharacteristic
39#16863449C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46Q8⋊(C67⋊C8)C2211yescharacteristic
47Q8⋊(C67⋊C8)C2211yescharacteristic
48#16863126C2211yescharacteristic
49#16863126C2211yescharacteristic
50#16863126C2211yescharacteristic
51#16863126C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863117C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56C2×C4×Q8×C67C2211yescharacteristic
57#446×C67—21no—
58#446×C67—21no—
59#446×C67C2211yescharacteristic
60#446×C67C2211yescharacteristic
61C8×Q8×C67C2211yescharacteristic
62C8×Q8×C67C2211yescharacteristic
63#432×C67—21no—
64#432×C67C2211yescharacteristic
65#431×C67—21no—
66#431×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761618C2×C411yescharacteristic
70C2×#11761618C2×C411yescharacteristic
71#14524043C2×C411yescharacteristic
72#14524043C2×C411yescharacteristic
73#14524035—21no—
74#14524022—21no—
75#14524022C2×C411yescharacteristic
76#14524022C2×C411yescharacteristic
77C4×Dic134—21no—
78C4×Dic134C2311yescharacteristic
79C4×Dic134C2311yescharacteristic
80#14524002—21no—
81#14523976C2×C411yescharacteristic
82#14523976C2×C411yescharacteristic
83#14523976C2×C411yescharacteristic
84#14523976C2×C411yescharacteristic
85C4⋊(C67⋊C8)—21no—
86C4⋊(C67⋊C8)C2×C411yescharacteristic
87C4⋊(C67⋊C8)C2×C411yescharacteristic
88C4⋊(C67⋊C8)C2311yescharacteristic
89C4⋊(C67⋊C8)C2311yescharacteristic
90Dic67⋊C8—21no—
91Dic67⋊C8—21no—
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523967C2×C411yescharacteristic
95#14523967C2×C411yescharacteristic
96#14523967C2×C411yescharacteristic
97#14523967C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×C411yescharacteristic
100#14523966C2×C411yescharacteristic
101#14523966C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106#14523953—21no—
107#14523953—21no—
108C4×C67⋊C8—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C22×Q8×C67C2×C411yescharacteristic
116C4×Q8×C67—21no—
117C4×Q8×C67—21no—
118C4×Q8×C67C2×C411yescharacteristic
119C4×Q8×C67C2×C411yescharacteristic
120C4×Q8×C67C2311yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C2×C4⋊C4×C67—21no—
123C2×C4⋊C4×C67C2×C411yescharacteristic
124C2×C42×C67—21no—
125C2×C42×C67C2311yescharacteristic
126C4⋊C8×C67—21no—
127C4⋊C8×C67—21no—
128C4⋊C8×C67—21no—
129C4⋊C8×C67—21no—
130C4⋊C8×C67D411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67D411yescharacteristic
136C22⋊C8×C67D411yescharacteristic
137#97×C67—21no—
138#97×C67—21no—
139#97×C67—21no—
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#11761618—41no—
150#11761618—41no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8C22×C411yescharacteristic
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C4⋊Dic67—21no—
160C4⋊Dic67—21no—
161C4⋊Dic67—21no—
162C4⋊Dic67C22×C411yescharacteristic
163C4⋊Dic67C22×C411yescharacteristic
164Dic67⋊C4—21no—
165Dic67⋊C4—21no—
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168C22⋊Dic67—21no—
169C22⋊Dic67—21no—
170C4×Dic67—21no—
171C4×Dic67—21no—
172C2×Q8×C67—21no—
173C2×Q8×C67—22no—
174C2×Q8×C67—41no—
175C2×Q8×C67C22×C411yescharacteristic
176C2×Q8×C67C22×C411yescharacteristic
177C22×C4×C67—21no—
178C22×C4×C67—21no—
179C22×C4×C67C22×C411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67#4111yescharacteristic
182C2×C8×C67—21no—
183C2×C8×C67—21no—
184C2×C8×C67C2×D411yescharacteristic
185C2×C8×C67C2×D411yescharacteristic
186C2×C8×C67C2×D411yescharacteristic
187C2×C8×C67C2×D411yescharacteristic
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67C22×C411yescharacteristic
194C4⋊C4×C67C22×C411yescharacteristic
195C42×C67—21no—
196C42×C67—21no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67C22×C411yescharacteristic
200C42×C67C2×D411yescharacteristic
201C42×C67#4111yescharacteristic
202C42×C67C2411yescharacteristic
203C23×C67C4×D411yescharacteristic
204Q8×C67—21no—
205Q8×C67—21no—
206Q8×C67—21no—
207Q8×C67—42no—
208Q8×C67#13112yes—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—41no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67C4×D411yescharacteristic
221C2×C4×C67C4×D411yescharacteristic
222C2×C4×C67C4×D411yescharacteristic
223C2×C4×C67C23×C411yescharacteristic
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×Dic67—21no—
227C2×Dic67—21no—
228C2×Dic67—21no—
229C2×Dic67—21no—
230Dic134—21no—
231Dic134—21no—
232Dic134—21no—
233Dic134—21no—
234Dic134—42no—
235C8×C67—21no—
236C8×C67—21no—
237C8×C67—21no—
238C8×C67—21no—
239C8×C67—41no—
240C8×C67—41no—
241C67⋊C8—21no—
242C67⋊C8—21no—
243C67⋊C8—21no—
244C67⋊C8—21no—
245C67⋊C8—41no—
246C67⋊C8—41no—
247C67⋊C8—41no—
248C67⋊C8—41no—
249C22×C67—21no—
250C22×C67—21no—
251C22×C67—21no—
252C22×C67C2×C4×D411yescharacteristic
253C4×C67—21no—
254C4×C67—22no—
255C4×C67—21no—
256C4×C67—21no—
257C4×C67—21no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67C2×#13111yescharacteristic
262C4×C67#56811yescharacteristic
263C4×C67#57811yescharacteristic
264C4×C67#57811yescharacteristic
265Dic67—41no—
266Dic67—41no—
267Dic67—41no—
268Dic67—41no—
269#18032—671nomaximal, Sylow
270C2×C67—41no—
271C2×C67#293311yescharacteristic
272C2×C67#295011yescharacteristic
273C2×C67#295511yescharacteristic
274#3104—671no—
275#2986—671no—
276#2984D6711yescharacteristic
277#1593—671no—
278#1591—671no—
279#1587—671no—
280#1587—671no—
281#1584—671no—
282#1584—671no—
283#1579—671no—
284C8×Q16—671no—
285#1572—671no—
286#1497—671no—
287#1479—671no—
288#1454—671no—
289C67#1803211yescharacteristic, complemented, Sylow
290#533—671no—
291C2×C4×Q8C2×D6711yescharacteristic
292#484—671no—
293#484—671no—
294#477—671no—
295#477—671no—
296#462—671no—
297#462—671no—
298#451—671no—
299#451—671no—
300#446—21no—
301#446—21no—
302#446—671no—
303#446—671no—
304#446—1341no—
305#446C2×D6711yescharacteristic
306#446C2×D6711yescharacteristic
307C8×Q8—671no—
308C8×Q8—671no—
309C8×Q8—1341no—
310C8×Q8C2×D6711yescharacteristic
311C8×Q8C2×D6711yescharacteristic
312C4×Q16—671no—
313C4×Q16—671no—
314C4×Q16—671no—
315C4×Q16—671no—
316#433—1341no—
317#433—1341no—
318#432—21no—
319#432C2×D6711yescharacteristic
320#431—21no—
321#431—671no—
322#431—671no—
323#431—671no—
324#431C2×D6711yescharacteristic
325#335—671no—
326#335—671no—
327#334—671no—
328#334—671no—
329Q8⋊C8—671no—
330Q8⋊C8—671no—
331Q8⋊C8—671no—
332Q8⋊C8—671no—
333Q8⋊C8—671no—
334Q8⋊C8—671no—
335Q8⋊C8—671no—
336Q8⋊C8—671no—
337#322—671no—
338#322—671no—
339#322—671no—
340C82—671no—
341C22×Q8C4×D6711yescharacteristic
342C2×Q16—671no—
343C2×Q16—671no—
344C2×Q16—671no—
345C2×Q16—671no—
346C4⋊Q8—671no—
347C4⋊Q8—671no—
348#125—1341no—
349C22⋊Q8—671no—
350C22⋊Q8—671no—
351C22⋊Q8—1341no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—671no—
355C4×Q8—671no—
356C4×Q8—1341no—
357C4×Q8C4×D6711yescharacteristic
358C4×Q8C4×D6711yescharacteristic
359C4×Q8C22×D6711yescharacteristic
360C4×Q8C22×D6711yescharacteristic
361#117—1341no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C4×D6711yescharacteristic
364C2×C42—21no—
365C2×C42C22×D6711yescharacteristic
366#107—671no—
367#107—671no—
368#107—671no—
369#107—671no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—671no—
375C4⋊C8—671no—
376C4⋊C8—671no—
377C4⋊C8—671no—
378C4⋊C8—1341no—
379C4⋊C8—1341no—
380C4⋊C8—1341no—
381C4⋊C8—1341no—
382C4⋊C8—1341no—
383C4⋊C8C22⋊D6711yescharacteristic
384C4⋊C8C22⋊D6711yescharacteristic
385C4⋊C8C22×D6711yescharacteristic
386C4⋊C8C22×D6711yescharacteristic
387#103—671no—
388#103—671no—
389#103—671no—
390#103—671no—
391#103—671no—
392#103—671no—
393#103—671no—
394#103—671no—
395C22⋊C8—21no—
396C22⋊C8—671no—
397C22⋊C8—671no—
398C22⋊C8—1341no—
399C22⋊C8—1341no—
400C22⋊C8C22⋊D6711yescharacteristic
401C22⋊C8C22⋊D6711yescharacteristic
402#97—21no—
403#97—21no—
404#97—21no—
405#97—1341no—
406#97—1341no—
407C4×C8—21no—
408C4×C8—21no—
409C4×C8—671no—
410C4×C8—671no—
411C4×C8—671no—
412C4×C8—671no—
413C4×C8—1341no—
414C4×C8—1341no—
415C4×C8—1341no—
416C4×C8C22×D6711yescharacteristic
417C4×C8C22×D6711yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—671no—
422C2×Q8—671no—
423C2×Q8—1341no—
424C2×Q8C2×C4×D6711yescharacteristic
425C2×Q8C2×C4×D6711yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×C4×D6711yescharacteristic
429C22×C4C2×C22⋊D6711yescharacteristic
430C22×C4#1176162811yescharacteristic
431Q16—2681no—
432Q16—2681no—
433Q16—1342no—
434Q16—1342no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—671no—
438C2×C8—671no—
439C2×C8—671no—
440C2×C8—671no—
441C2×C8—1341no—
442C2×C8—1341no—
443C2×C8—1341no—
444C2×C8—1341no—
445C2×C8C2×C22⋊D6711yescharacteristic
446C2×C8C2×C22⋊D6711yescharacteristic
447C2×C8C2×C22⋊D6711yescharacteristic
448C2×C8C2×C22⋊D6711yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—671no—
455C4⋊C4—671no—
456C4⋊C4—1341no—
457C4⋊C4—1341no—
458C4⋊C4—1341no—
459C4⋊C4—1341no—
460C4⋊C4—1341no—
461C4⋊C4—1341no—
462C4⋊C4—1341no—
463C4⋊C4C2×C4×D6711yescharacteristic
464C4⋊C4C2×C4×D6711yescharacteristic
465C22⋊C4—1341no—
466C22⋊C4—1341no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1341no—
472C42—1341no—
473C42C2×C4×D6711yescharacteristic
474C42C2×C22⋊D6711yescharacteristic
475C42#1176162811yescharacteristic
476C42C23×D6711yescharacteristic
477C23C4×C22⋊D6711yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—42no—
482Q8—2682no—
483Q8—1341no—
484Q8—1341no—
485Q8—1341no—
486Q8—1341no—
487Q8#1452405112yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1341no—
500C2×C4—1341no—
501C2×C4—1341no—
502C2×C4—1341no—
503C2×C4C4×C22⋊D6711yescharacteristic
504C2×C4C4×C22⋊D6711yescharacteristic
505C2×C4C4×C22⋊D6711yescharacteristic
506C2×C4C22×C4×D6711yescharacteristic
507C2×C4C22×C22⋊D6711yescharacteristic
508C2×C4C2×#1176162811yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—2681no—
516C8—2681no—
517C8—2681no—
518C8—2681no—
519C8—1341no—
520C8—1341no—
521C8—1341no—
522C8—1341no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×C4×C22⋊D6711yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—2681no—
536C4—2681no—
537C4—2681no—
538C4—2681no—
539C4C2×#1452405111yescharacteristic
540C4#1686416211yescharacteristic
541C4#1686421111yescharacteristic
542C4#1686421111yescharacteristic
543C2—41no—
544C2#2256893411yescharacteristic
545C2#2256907511yescharacteristic
546C2#2256911011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2984C2
socleC22×C67C2×C4×D4
prime radicalC23×C67C4×D4
cosocleC42×C67C24
hypercenter#2984D67
hypocenterC67#18032
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2984

Complements

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

NH ≅ G/Ntype
3. #22559116C2semidirect
4. #22559104C2semidirect
5. #22559080C2semidirect
6. #22559078C2semidirect
7. #22559059C2semidirect
8. #22559057C2semidirect
9. #22559026C2semidirect
10. C8×#11761618C2semidirect
289. C67#18032semidirect

Maximal subgroups

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

subgroupindex
#225702482
#225693102
#225591162
#225591042
#225590802
#225590782
#225590592
#225590572
#225590262
C8×#117616182
#225589902
#225586812
#225586042
#225585092
C67×#29842
#1803267