Finite Group Databasebeta

← #31797312

Subgroups — #31797312

Order 17536.

652 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#10587739C211yesmaximal, characteristic
2C2×#17984167C211yesmaximal, characteristic, complemented
3#22578354C211yesmaximal, characteristic, complemented
4#22578353C211yesmaximal, characteristic, complemented
5#22578353C211yesmaximal, characteristic, complemented
6#22578352C211yesmaximal, characteristic, complemented
7#22578352C211yesmaximal, characteristic, complemented
8#22578351C211yesmaximal, characteristic
9#22578350C211yesmaximal, characteristic, complemented
10#22578350C211yesmaximal, characteristic, complemented
11C4⋊C4×#10587739C211yesmaximal, characteristic, complemented
12C22⋊C4×#10587739C211yesmaximal, characteristic, complemented
13C22⋊C4×#10587739C211yesmaximal, characteristic, complemented
14C2×(C2×#10587739)⋊C4C211yesmaximal, characteristic, complemented
15#123×D137C211yesmaximal, characteristic
16C23×#10587739C2211yescharacteristic
17C23×#10587739C2211yescharacteristic
18C23×#10587739C2211yescharacteristic
19D4×#10587739—24no—
20#17984167—24no—
21C2×C4⋊#10587739C2211yescharacteristic
22C2×C4⋊#10587739C2211yescharacteristic
23C2×C4⋊#10587739C2211yescharacteristic
24C2×C4⋊#10587739C2211yescharacteristic
25C2×C4⋊#10587739C2211yescharacteristic
26C2×C22⋊#10587739C2211yescharacteristic
27C2×C22⋊#10587739C2211yescharacteristic
28C2×C22⋊#10587739C2211yescharacteristic
29C2×C22⋊#10587739C2211yescharacteristic
30C2×C22⋊#10587739C2211yescharacteristic
31C2×C4×#10587739C2211yescharacteristic
32C2×C4×#10587739C2211yescharacteristic
33C2×C4×#10587739C2211yescharacteristic
34C2×C4×#10587739C2211yescharacteristic
35C2×C4×#10587739C2211yescharacteristic
36(C2×#10587739)⋊C4C2211yescharacteristic
37(C2×#10587739)⋊C4C2211yescharacteristic
38(C2×#10587739)⋊C4C2211yescharacteristic
39(C2×#10587739)⋊C4C2211yescharacteristic
40(C2×#10587739)⋊C4C2211yescharacteristic
41(C2×#10587739)⋊C4C2211yescharacteristic
42(C2×#10587739)⋊C4C2211yescharacteristic
43(C2×#10587739)⋊C4C2211yescharacteristic
44(C2×#10587739)⋊C4C2211yescharacteristic
45(C2×#10587739)⋊C4C2211yescharacteristic
46C2×D4×D137C2211yescharacteristic
47C22×C4×D137C2211yescharacteristic
48#17984073C411yescharacteristic, complemented
49#17984073C411yescharacteristic, complemented
50#17984072C411yescharacteristic, complemented
51#17984071C411yescharacteristic, complemented
52#17984070C411yescharacteristic, complemented
53#17984070C411yescharacteristic, complemented
54#17984069C411yescharacteristic, complemented
55C4⋊C4×D137C2211yescharacteristic
56C4⋊C4×D137C2211yescharacteristic
57C22⋊C4×D137C2211yescharacteristic
58C22⋊C4×D137C2211yescharacteristic
59C22⋊C4×D137C2211yescharacteristic
60#123×C137C411yescharacteristic, complemented
61C22×#10587739—22no—
62C22×#10587739—22no—
63C22×#10587739—22no—
64C22×#10587739—22no—
65C22×#10587739—22no—
66C22×#10587739—22no—
67C22×#10587739—41no—
68C22×#10587739—41no—
69C22×#10587739—41no—
70C22×#10587739C2311yescharacteristic
71C22×#10587739C2311yescharacteristic
72C22×#10587739C2311yescharacteristic
73C22×#10587739C2311yescharacteristic
74C22×#10587739C2311yescharacteristic
75C22×#10587739C2311yescharacteristic
76C22×#10587739C2311yescharacteristic
77C22×#10587739C2311yescharacteristic
78C4⋊#10587739—22no—
79C4⋊#10587739—22no—
80C4⋊#10587739—22no—
81C4⋊#10587739—22no—
82C4⋊#10587739—41no—
83C22⋊#10587739—22no—
84C22⋊#10587739—22no—
85C22⋊#10587739—22no—
86C22⋊#10587739—41no—
87C22⋊#10587739D414yes—
88C4×#10587739—22no—
89C4×#10587739—22no—
90C4×#10587739—22no—
91C4×#10587739—22no—
92C4×#10587739—41no—
93C23×D137C2311yescharacteristic
94C23×D137C2311yescharacteristic
95C2×D548C2×C411yescharacteristic
96C2×C22⋊D137C2×C411yescharacteristic
97C2×C22⋊D137C2×C411yescharacteristic
98D4×D137—22no—
99D4×D137—22no—
100C22×Dic137C2×C411yescharacteristic, complemented
101C2×C4×D137—22no—
102C2×C4×D137—22no—
103C2×C4×D137C2×C411yescharacteristic, complemented
104C2×C4×D137C2×C411yescharacteristic, complemented
105C2×C4×D137C2311yescharacteristic
106C2×C4×D137C2311yescharacteristic
107C2×C4×D137C2311yescharacteristic
108C2×C4×D137C2311yescharacteristic
109C2×C4×D137C2311yescharacteristic
110C4⋊Dic137C2×C411yescharacteristic, complemented
111C4⋊Dic137C2×C411yescharacteristic, complemented
112#14555489C2×C411yescharacteristic, complemented
113#14555489C2×C411yescharacteristic, complemented
114#14555489C2×C411yescharacteristic, complemented
115#14555489C2×C411yescharacteristic, complemented
116C22⋊Dic137C2×C411yescharacteristic, complemented
117C22⋊Dic137C2×C411yescharacteristic, complemented
118C22⋊Dic137C2×C411yescharacteristic, complemented
119#14555487C2×C411yescharacteristic, complemented
120#14555487C2×C411yescharacteristic, complemented
121#14555487C2×C411yescharacteristic, complemented
122#14555487C2×C411yescharacteristic, complemented
123#14555487C2×C411yescharacteristic, complemented
124#14555487C2×C411yescharacteristic, complemented
125C2×D4×C137C2×C411yescharacteristic
126C22×C4×C137C2×C411yescharacteristic, complemented
127C4⋊C4×C137C2×C411yescharacteristic, complemented
128C4⋊C4×C137C2×C411yescharacteristic, complemented
129C22⋊C4×C137C2×C411yescharacteristic, complemented
130C22⋊C4×C137C2×C411yescharacteristic, complemented
131C22⋊C4×C137C2×C411yescharacteristic, complemented
132C2×#10587739—21no—
133C2×#10587739—21no—
134C2×#10587739—21no—
135C2×#10587739—21no—
136C2×#10587739—21no—
137C2×#10587739—21no—
138C2×#10587739—21no—
139C2×#10587739—21no—
140C2×#10587739—21no—
141C2×#10587739—21no—
142C2×#10587739—21no—
143C2×#10587739—21no—
144C2×#10587739—21no—
145C2×#10587739—21no—
146C2×#10587739—21no—
147C2×#10587739—21no—
148C2×#10587739—21no—
149C2×#10587739—21no—
150C2×#10587739—21no—
151C2×#10587739—21no—
152C2×#10587739—44no—
153C2×#10587739—44no—
154C2×#10587739—44no—
155C2×#10587739C2×D412yes—
156C2×#10587739C2×D412yes—
157C2×#10587739#4112yes—
158C2×#10587739#4112yes—
159C22×D137—21no—
160C22×D137—21no—
161C22×D137—21no—
162C22×D137—21no—
163C22×D137—21no—
164C22×D137—21no—
165C22×D137—21no—
166C22×D137—21no—
167C22×D137—21no—
168C22×D137—21no—
169C22×D137C22×C411yescharacteristic
170C22×D137C22×C411yescharacteristic
171C22×D137C2×D412yes—
172C22×D137#4112yes—
173C22×D137C2411yescharacteristic
174D548—22no—
175C22⋊D137—22no—
176C22⋊D137—22no—
177C2×Dic137—22no—
178C2×Dic137C22×C411yescharacteristic
179C2×Dic137C22×C411yescharacteristic
180C2×Dic137C22×C411yescharacteristic
181C2×Dic137C22×C411yescharacteristic
182C2×Dic137C22×C411yescharacteristic
183C4×D137—22no—
184C4×D137—22no—
185C4×D137—21no—
186C4×D137—21no—
187C4×D137—21no—
188C4×D137—21no—
189C4×D137—21no—
190C4×D137—21no—
191C4×D137—21no—
192C4×D137—21no—
193C4×D137—21no—
194C4×D137—21no—
195C23×C137C22×C411yescharacteristic
196C23×C137C22×C411yescharacteristic
197D4×C137—22no—
198C2×C4×C137—22no—
199C2×C4×C137C22×C411yescharacteristic
200C2×C4×C137C22×C411yescharacteristic
201C2×C4×C137C22×C411yescharacteristic
202C2×C4×C137C22×C411yescharacteristic
203C2×C4×C137C22×C411yescharacteristic
204C22×C137—21no—
205C22×C137—21no—
206C22×C137—21no—
207C22×C137—21no—
208C22×C137—21no—
209C22×C137C4×D412yes—
210C22×C137C23×C411yescharacteristic
211C2×D137—21no—
212C2×D137—21no—
213C2×D137—22no—
214C2×D137—22no—
215C2×D137—22no—
216C2×D137—22no—
217C2×D137—21no—
218C2×D137—21no—
219C2×D137—21no—
220C2×D137—41no—
221C2×D137—41no—
222C2×D137—41no—
223C2×D137—41no—
224C2×D137C4×D412yes—
225C2×D137C22×D411yescharacteristic
226C2×D137C2×#4111yescharacteristic
227C2×D137C2×#4111yescharacteristic
228C2×D137C2×#4111yescharacteristic
229C2×D137H(5,𝔽2)11yescharacteristic
230C2×D1372−1+411yescharacteristic
231#10587739—22no—
232#10587739—22no—
233#10587739—22no—
234#10587739—22no—
235#10587739—41no—
236#10587739—41no—
237#10587739—41no—
238#10587739—41no—
239C4×C137—21no—
240C4×C137—21no—
241C4×C137—21no—
242C4×C137—21no—
243C4×C137—21no—
244Dic137—21no—
245Dic137—21no—
246Dic137—21no—
247Dic137—21no—
248Dic137—21no—
249C2×C137—22no—
250C2×C137—41no—
251C2×C137C2×C4×D411yescharacteristic
252C2×C137C4×#4111yescharacteristic
253C2×C137#52011yescharacteristic
254D137—22no—
255D137—41no—
256D137#54611yescharacteristic
257D137#54711yescharacteristic
258D137#55111yescharacteristic
259D137#55211yescharacteristic
260C137#235911yescharacteristic, complemented, Sylow
261#2359—1371nomaximal, Sylow
262C2×#123—1371no—
263C2×C4×D4—1371no—
264C2×#117—1371no—
265#389—1371no—
266#388—1371no—
267#388—1371no—
268#387—1371no—
269#387—1371no—
270#386—1371no—
271#385—1371no—
272#385—1371no—
273C4×C4⋊C4—1371no—
274C4×C22⋊C4—1371no—
275C4×C22⋊C4—1371no—
276C2×#95—1371no—
277C22×D4—1371no—
278C23×C4—1371no—
279C23×C4—1371no—
280C23×C4—1371no—
281C23×C4—1371no—
282#123—1371no—
283#123—1371no—
284#123—1371no—
285#123—1371no—
286#123—1371no—
287#123—1371no—
288#123—1371no—
289#123#1058773911yescharacteristic, complemented
290C4×D4—2744no—
291#117—2744no—
292C2×C4⋊C4—1371no—
293C2×C4⋊C4—1371no—
294C2×C4⋊C4—1371no—
295C2×C4⋊C4—1371no—
296C2×C4⋊C4—1371no—
297C2×C4⋊C4—1371no—
298C2×C4⋊C4—1371no—
299C2×C22⋊C4—1371no—
300C2×C22⋊C4—1371no—
301C2×C22⋊C4—1371no—
302C2×C22⋊C4—1371no—
303C2×C22⋊C4—1371no—
304C2×C22⋊C4—1371no—
305C2×C22⋊C4—1371no—
306C2×C22⋊C4—1371no—
307C2×C42—1371no—
308C2×C42—1371no—
309C2×C42—1371no—
310C2×C42—1371no—
311C2×C42—1371no—
312#95—1371no—
313#95—1371no—
314#95—1371no—
315#95—1371no—
316#95—1371no—
317#95—1371no—
318#95—1371no—
319#95—1371no—
320#95—1371no—
321#95—1371no—
322C24—1371no—
323C24—1371no—
324C2×D4—2742no—
325C2×D4—2742no—
326C2×D4—1371no—
327C2×D4—1371no—
328C2×D4—1371no—
329C2×D4C2×#1058773911yescharacteristic
330C22×C4—2742no—
331C22×C4—2742no—
332C22×C4—2742no—
333C22×C4—2742no—
334C22×C4—2742no—
335C22×C4—2742no—
336C22×C4—2742no—
337C22×C4—2742no—
338C22×C4—5481no—
339C22×C4—5481no—
340C22×C4—5481no—
341C22×C4—1371no—
342C22×C4—1371no—
343C22×C4—1371no—
344C22×C4—1371no—
345C22×C4—1371no—
346C22×C4—1371no—
347C22×C4—1371no—
348C22×C4—1371no—
349C22×C4—1371no—
350C22×C4—1371no—
351C22×C4—1371no—
352C22×C4—1371no—
353C22×C4—1371no—
354C22×C4—1371no—
355C22×C4—1371no—
356C22×C4—1371no—
357C22×C4C2×#1058773911yescharacteristic, complemented
358C4⋊C4—2742no—
359C4⋊C4—2742no—
360C4⋊C4—2742no—
361C4⋊C4—2742no—
362C4⋊C4—5481no—
363C4⋊C4—1371no—
364C4⋊C4—1371no—
365C4⋊C4—1371no—
366C4⋊C4—1371no—
367C4⋊C4—1371no—
368C4⋊C4—1371no—
369C4⋊C4C2×#1058773911yescharacteristic, complemented
370C4⋊C4C2×#1058773911yescharacteristic, complemented
371C22⋊C4—2742no—
372C22⋊C4—2742no—
373C22⋊C4—2742no—
374C22⋊C4—5481no—
375C22⋊C4—1374no—
376C22⋊C4—1371no—
377C22⋊C4—1371no—
378C22⋊C4—1371no—
379C22⋊C4—1371no—
380C22⋊C4—1371no—
381C22⋊C4—1371no—
382C22⋊C4—1371no—
383C22⋊C4—1371no—
384C22⋊C4—1371no—
385C22⋊C4C2×#1058773911yescharacteristic, complemented
386C22⋊C4C2×#1058773911yescharacteristic, complemented
387C22⋊C4C2×#1058773911yescharacteristic, complemented
388C42—2742no—
389C42—2742no—
390C42—2742no—
391C42—2742no—
392C42—5481no—
393C23—2741no—
394C23—2741no—
395C23—2741no—
396C23—2741no—
397C23—2741no—
398C23—2741no—
399C23—2741no—
400C23—2741no—
401C23—2741no—
402C23—2741no—
403C23—1371no—
404C23—1372no—
405C23—1372no—
406C23—1371no—
407C23—1371no—
408C23C22×#1058773911yescharacteristic
409C23C22×#1058773911yescharacteristic
410D4—22no—
411D4—2742no—
412D4—2742no—
413D4—2742no—
414C2×C4—22no—
415C2×C4—2741no—
416C2×C4—2741no—
417C2×C4—2741no—
418C2×C4—2741no—
419C2×C4—2741no—
420C2×C4—2741no—
421C2×C4—2742no—
422C2×C4—2742no—
423C2×C4—2741no—
424C2×C4—2741no—
425C2×C4—2741no—
426C2×C4—2741no—
427C2×C4—2741no—
428C2×C4—2741no—
429C2×C4—2741no—
430C2×C4—2741no—
431C2×C4—2741no—
432C2×C4—2741no—
433C2×C4—2741no—
434C2×C4—2741no—
435C2×C4—2742no—
436C2×C4—2741no—
437C2×C4—2741no—
438C2×C4—2741no—
439C2×C4—2741no—
440C2×C4—2741no—
441C2×C4—2741no—
442C2×C4—2741no—
443C2×C4—2741no—
444C2×C4—2741no—
445C2×C4—2741no—
446C2×C4—2741no—
447C2×C4—2741no—
448C2×C4—5484no—
449C2×C4—5484no—
450C2×C4—5484no—
451C2×C4—1372no—
452C2×C4—1372no—
453C2×C4—1372no—
454C2×C4—1372no—
455C2×C4—1371no—
456C2×C4—1371no—
457C2×C4—1371no—
458C2×C4—1371no—
459C2×C4—1371no—
460C2×C4C22×#1058773911yescharacteristic
461C2×C4C22×#1058773911yescharacteristic
462C2×C4C22×#1058773911yescharacteristic
463C2×C4C22×#1058773911yescharacteristic
464C2×C4C22×#1058773911yescharacteristic
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—2742no—
471C22—2742no—
472C22—2742no—
473C22—2742no—
474C22—2741no—
475C22—2741no—
476C22—2741no—
477C22—2741no—
478C22—2741no—
479C22—5481no—
480C22—5481no—
481C22—5481no—
482C22—5481no—
483C22—1372no—
484C22—1371no—
485C22—1371no—
486C22—1371no—
487C22—1371no—
488C22—1371no—
489C22—1371no—
490C22D4×#1058773912yes—
491C22C23×#1058773911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—2742no—
498C4—2742no—
499C4—2742no—
500C4—2742no—
501C4—2741no—
502C4—2741no—
503C4—2741no—
504C4—2741no—
505C4—2741no—
506C4—5481no—
507C4—5481no—
508C4—5481no—
509C4—5481no—
510C2—22no—
511C2—41no—
512C2—2742no—
513C2—5481no—
514C2—1371no—
515C2—1371no—
516C2—1371no—
517C2—1371no—
518C2C2×D4×#1058773911yescharacteristic
519C2#41×#1058773911yescharacteristic
520C2#2257847211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
Fitting#123×C137C4
prime radicalC2×D4×D137C22
cosocleC22×D137C24
hypercenter#123#10587739
hypocenterC137#2359
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C2×#17984167C2semidirect
3. #22578354C2semidirect
4. #22578353C2semidirect
5. #22578353C2semidirect
6. #22578352C2semidirect
7. #22578352C2semidirect
9. #22578350C2semidirect
10. #22578350C2semidirect
11. C4⋊C4×#10587739C2semidirect
12. C22⋊C4×#10587739C2semidirect
13. C22⋊C4×#10587739C2semidirect
14. C2×(C2×#10587739)⋊C4C2semidirect
48. #17984073C4semidirect
49. #17984073C4semidirect
50. #17984072C4semidirect
51. #17984071C4semidirect
52. #17984070C4semidirect
53. #17984070C4semidirect
54. #17984069C4semidirect
60. #123×C137C4semidirect
100. C22×Dic137C2×C4semidirect
103. C2×C4×D137C2×C4semidirect
104. C2×C4×D137C2×C4semidirect
110. C4⋊Dic137C2×C4semidirect
111. C4⋊Dic137C2×C4semidirect
112. #14555489C2×C4semidirect
113. #14555489C2×C4semidirect
114. #14555489C2×C4semidirect
115. #14555489C2×C4semidirect
116. C22⋊Dic137C2×C4semidirect
117. C22⋊Dic137C2×C4semidirect
118. C22⋊Dic137C2×C4semidirect
119. #14555487C2×C4semidirect
120. #14555487C2×C4semidirect
121. #14555487C2×C4semidirect
122. #14555487C2×C4semidirect
123. #14555487C2×C4semidirect
124. #14555487C2×C4semidirect
126. C22×C4×C137C2×C4semidirect
127. C4⋊C4×C137C2×C4semidirect
128. C4⋊C4×C137C2×C4semidirect
129. C22⋊C4×C137C2×C4semidirect
130. C22⋊C4×C137C2×C4semidirect
131. C22⋊C4×C137C2×C4semidirect
260. C137#2359semidirect
289. #123#10587739semidirect
357. C22×C4C2×#10587739semidirect
369. C4⋊C4C2×#10587739semidirect
370. C4⋊C4C2×#10587739semidirect
385. C22⋊C4C2×#10587739semidirect
386. C22⋊C4C2×#10587739semidirect
387. C22⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#105877392
C2×#179841672
#225783542
#225783532
#225783522
#225783512
#225783502
C4⋊C4×#105877392
C22⋊C4×#105877392
C2×(C2×#10587739)⋊C42
#123×D1372
#2359137