Finite Group Databasebeta

← #27158259

Subgroups — #27158259

Order 12416.

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×#87604C211yesmaximal, characteristic
2C2×#15690889C211yesmaximal, characteristic, complemented
3#20261686C211yesmaximal, characteristic, complemented
4#20261685C211yesmaximal, characteristic, complemented
5#20261685C211yesmaximal, characteristic, complemented
6#20261684C211yesmaximal, characteristic, complemented
7#20261684C211yesmaximal, characteristic, complemented
8#20261683C211yesmaximal, characteristic
9#20261682C211yesmaximal, characteristic, complemented
10#20261682C211yesmaximal, characteristic, complemented
11C4⋊C4×#87604C211yesmaximal, characteristic, complemented
12C22⋊C4×#87604C211yesmaximal, characteristic, complemented
13C22⋊C4×#87604C211yesmaximal, characteristic, complemented
14C2×(C2×#87604)⋊C4C211yesmaximal, characteristic, complemented
15#123×D97C211yesmaximal, characteristic
16C23×#87604C2211yescharacteristic
17C23×#87604C2211yescharacteristic
18C23×#87604C2211yescharacteristic
19D4×#87604—24no—
20#15690889—24no—
21C2×C4⋊#87604C2211yescharacteristic
22C2×C4⋊#87604C2211yescharacteristic
23C2×C4⋊#87604C2211yescharacteristic
24C2×C4⋊#87604C2211yescharacteristic
25C2×C4⋊#87604C2211yescharacteristic
26C2×C22⋊#87604C2211yescharacteristic
27C2×C22⋊#87604C2211yescharacteristic
28C2×C22⋊#87604C2211yescharacteristic
29C2×C22⋊#87604C2211yescharacteristic
30C2×C22⋊#87604C2211yescharacteristic
31C2×C4×#87604C2211yescharacteristic
32C2×C4×#87604C2211yescharacteristic
33C2×C4×#87604C2211yescharacteristic
34C2×C4×#87604C2211yescharacteristic
35C2×C4×#87604C2211yescharacteristic
36(C2×#87604)⋊C4C2211yescharacteristic
37(C2×#87604)⋊C4C2211yescharacteristic
38(C2×#87604)⋊C4C2211yescharacteristic
39(C2×#87604)⋊C4C2211yescharacteristic
40(C2×#87604)⋊C4C2211yescharacteristic
41(C2×#87604)⋊C4C2211yescharacteristic
42(C2×#87604)⋊C4C2211yescharacteristic
43(C2×#87604)⋊C4C2211yescharacteristic
44(C2×#87604)⋊C4C2211yescharacteristic
45(C2×#87604)⋊C4C2211yescharacteristic
46C2×D4×D97C2211yescharacteristic
47C22×C4×D97C2211yescharacteristic
48#15690795C411yescharacteristic, complemented
49#15690795C411yescharacteristic, complemented
50#15690794C411yescharacteristic, complemented
51#15690793C411yescharacteristic, complemented
52#15690792C411yescharacteristic, complemented
53#15690792C411yescharacteristic, complemented
54#15690791C411yescharacteristic, complemented
55C4⋊C4×D97C2211yescharacteristic
56C4⋊C4×D97C2211yescharacteristic
57C22⋊C4×D97C2211yescharacteristic
58C22⋊C4×D97C2211yescharacteristic
59C22⋊C4×D97C2211yescharacteristic
60#123×C97C411yescharacteristic, complemented
61C22×#87604—22no—
62C22×#87604—22no—
63C22×#87604—22no—
64C22×#87604—22no—
65C22×#87604—22no—
66C22×#87604—22no—
67C22×#87604—41no—
68C22×#87604—41no—
69C22×#87604—41no—
70C22×#87604C2311yescharacteristic
71C22×#87604C2311yescharacteristic
72C22×#87604C2311yescharacteristic
73C22×#87604C2311yescharacteristic
74C22×#87604C2311yescharacteristic
75C22×#87604C2311yescharacteristic
76C22×#87604C2311yescharacteristic
77C22×#87604C2311yescharacteristic
78C4⋊#87604—22no—
79C4⋊#87604—22no—
80C4⋊#87604—22no—
81C4⋊#87604—22no—
82C4⋊#87604—41no—
83C22⋊#87604—22no—
84C22⋊#87604—22no—
85C22⋊#87604—22no—
86C22⋊#87604—41no—
87C22⋊#87604D414yes—
88C4×#87604—22no—
89C4×#87604—22no—
90C4×#87604—22no—
91C4×#87604—22no—
92C4×#87604—41no—
93C23×D97C2311yescharacteristic
94C23×D97C2311yescharacteristic
95C2×D388C2×C411yescharacteristic
96C2×C22⋊D97C2×C411yescharacteristic
97C2×C22⋊D97C2×C411yescharacteristic
98D4×D97—22no—
99D4×D97—22no—
100C22×Dic97C2×C411yescharacteristic, complemented
101C2×C4×D97—22no—
102C2×C4×D97—22no—
103C2×C4×D97C2×C411yescharacteristic, complemented
104C2×C4×D97C2×C411yescharacteristic, complemented
105C2×C4×D97C2311yescharacteristic
106C2×C4×D97C2311yescharacteristic
107C2×C4×D97C2311yescharacteristic
108C2×C4×D97C2311yescharacteristic
109C2×C4×D97C2311yescharacteristic
110C4⋊Dic97C2×C411yescharacteristic, complemented
111C4⋊Dic97C2×C411yescharacteristic, complemented
112#13095490C2×C411yescharacteristic, complemented
113#13095490C2×C411yescharacteristic, complemented
114#13095490C2×C411yescharacteristic, complemented
115#13095490C2×C411yescharacteristic, complemented
116C22⋊Dic97C2×C411yescharacteristic, complemented
117C22⋊Dic97C2×C411yescharacteristic, complemented
118C22⋊Dic97C2×C411yescharacteristic, complemented
119#13095488C2×C411yescharacteristic, complemented
120#13095488C2×C411yescharacteristic, complemented
121#13095488C2×C411yescharacteristic, complemented
122#13095488C2×C411yescharacteristic, complemented
123#13095488C2×C411yescharacteristic, complemented
124#13095488C2×C411yescharacteristic, complemented
125C2×D4×C97C2×C411yescharacteristic
126C22×C4×C97C2×C411yescharacteristic, complemented
127C4⋊C4×C97C2×C411yescharacteristic, complemented
128C4⋊C4×C97C2×C411yescharacteristic, complemented
129C22⋊C4×C97C2×C411yescharacteristic, complemented
130C22⋊C4×C97C2×C411yescharacteristic, complemented
131C22⋊C4×C97C2×C411yescharacteristic, complemented
132C23×C97C22×C411yescharacteristic
133C23×C97C22×C411yescharacteristic
134C22×D97—21no—
135C22×D97—21no—
136C22×D97—21no—
137C22×D97—21no—
138C22×D97—21no—
139C22×D97—21no—
140C22×D97—21no—
141C22×D97—21no—
142C22×D97—21no—
143C22×D97—21no—
144C22×D97C22×C411yescharacteristic
145C22×D97C22×C411yescharacteristic
146C22×D97C2×D412yes—
147C22×D97#4112yes—
148C22×D97C2411yescharacteristic
149C2×#87604—21no—
150C2×#87604—21no—
151C2×#87604—21no—
152C2×#87604—21no—
153C2×#87604—21no—
154C2×#87604—21no—
155C2×#87604—21no—
156C2×#87604—21no—
157C2×#87604—21no—
158C2×#87604—21no—
159C2×#87604—21no—
160C2×#87604—21no—
161C2×#87604—21no—
162C2×#87604—21no—
163C2×#87604—21no—
164C2×#87604—21no—
165C2×#87604—21no—
166C2×#87604—21no—
167C2×#87604—21no—
168C2×#87604—21no—
169C2×#87604—44no—
170C2×#87604—44no—
171C2×#87604—44no—
172C2×#87604C2×D412yes—
173C2×#87604C2×D412yes—
174C2×#87604#4112yes—
175C2×#87604#4112yes—
176D4×C97—22no—
177C2×C4×C97—22no—
178C2×C4×C97C22×C411yescharacteristic
179C2×C4×C97C22×C411yescharacteristic
180C2×C4×C97C22×C411yescharacteristic
181C2×C4×C97C22×C411yescharacteristic
182C2×C4×C97C22×C411yescharacteristic
183C22⋊D97—22no—
184C22⋊D97—22no—
185C2×Dic97—22no—
186C2×Dic97C22×C411yescharacteristic
187C2×Dic97C22×C411yescharacteristic
188C2×Dic97C22×C411yescharacteristic
189C2×Dic97C22×C411yescharacteristic
190C2×Dic97C22×C411yescharacteristic
191D388—22no—
192C4×D97—22no—
193C4×D97—22no—
194C4×D97—21no—
195C4×D97—21no—
196C4×D97—21no—
197C4×D97—21no—
198C4×D97—21no—
199C4×D97—21no—
200C4×D97—21no—
201C4×D97—21no—
202C4×D97—21no—
203C4×D97—21no—
204C22×C97—21no—
205C22×C97—21no—
206C22×C97—21no—
207C22×C97—21no—
208C22×C97—21no—
209C22×C97C4×D412yes—
210C22×C97C23×C411yescharacteristic
211C2×D97—21no—
212C2×D97—21no—
213C2×D97—22no—
214C2×D97—22no—
215C2×D97—22no—
216C2×D97—22no—
217C2×D97—21no—
218C2×D97—21no—
219C2×D97—21no—
220C2×D97—41no—
221C2×D97—41no—
222C2×D97—41no—
223C2×D97—41no—
224C2×D97C4×D412yes—
225C2×D97C22×D411yescharacteristic
226C2×D97C2×#4111yescharacteristic
227C2×D97C2×#4111yescharacteristic
228C2×D97C2×#4111yescharacteristic
229C2×D97H(5,𝔽2)11yescharacteristic
230C2×D972−1+411yescharacteristic
231#87604—22no—
232#87604—22no—
233#87604—22no—
234#87604—22no—
235#87604—41no—
236#87604—41no—
237#87604—41no—
238#87604—41no—
239C4×C97—21no—
240C4×C97—21no—
241C4×C97—21no—
242C4×C97—21no—
243C4×C97—21no—
244Dic97—21no—
245Dic97—21no—
246Dic97—21no—
247Dic97—21no—
248Dic97—21no—
249C2×C97—22no—
250C2×C97—41no—
251C2×C97C2×C4×D411yescharacteristic
252C2×C97C4×#4111yescharacteristic
253C2×C97#52011yescharacteristic
254D97—22no—
255D97—41no—
256D97#54611yescharacteristic
257D97#54711yescharacteristic
258D97#55111yescharacteristic
259D97#55211yescharacteristic
260#2359—971nomaximal, Sylow
261C97#235911yescharacteristic, complemented, Sylow
262C2×#123—971no—
263C2×C4×D4—971no—
264C2×#117—971no—
265#389—971no—
266#388—971no—
267#388—971no—
268#387—971no—
269#387—971no—
270#386—971no—
271#385—971no—
272#385—971no—
273C4×C4⋊C4—971no—
274C4×C22⋊C4—971no—
275C4×C22⋊C4—971no—
276C2×#95—971no—
277C22×D4—971no—
278C23×C4—971no—
279C23×C4—971no—
280C23×C4—971no—
281C23×C4—971no—
282#123—971no—
283#123—971no—
284#123—971no—
285#123—971no—
286#123—971no—
287#123—971no—
288#123—971no—
289#123#8760411yescharacteristic, complemented
290C4×D4—1944no—
291#117—1944no—
292C2×C4⋊C4—971no—
293C2×C4⋊C4—971no—
294C2×C4⋊C4—971no—
295C2×C4⋊C4—971no—
296C2×C4⋊C4—971no—
297C2×C4⋊C4—971no—
298C2×C4⋊C4—971no—
299C2×C22⋊C4—971no—
300C2×C22⋊C4—971no—
301C2×C22⋊C4—971no—
302C2×C22⋊C4—971no—
303C2×C22⋊C4—971no—
304C2×C22⋊C4—971no—
305C2×C22⋊C4—971no—
306C2×C22⋊C4—971no—
307C2×C42—971no—
308C2×C42—971no—
309C2×C42—971no—
310C2×C42—971no—
311C2×C42—971no—
312#95—971no—
313#95—971no—
314#95—971no—
315#95—971no—
316#95—971no—
317#95—971no—
318#95—971no—
319#95—971no—
320#95—971no—
321#95—971no—
322C24—971no—
323C24—971no—
324C2×D4—971no—
325C2×D4—971no—
326C2×D4—971no—
327C2×D4—1942no—
328C2×D4—1942no—
329C2×D4C2×#8760411yescharacteristic
330C22×C4—971no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—971no—
334C22×C4—971no—
335C22×C4—971no—
336C22×C4—971no—
337C22×C4—971no—
338C22×C4—971no—
339C22×C4—971no—
340C22×C4—971no—
341C22×C4—971no—
342C22×C4—971no—
343C22×C4—971no—
344C22×C4—971no—
345C22×C4—971no—
346C22×C4—3881no—
347C22×C4—3881no—
348C22×C4—3881no—
349C22×C4—1942no—
350C22×C4—1942no—
351C22×C4—1942no—
352C22×C4—1942no—
353C22×C4—1942no—
354C22×C4—1942no—
355C22×C4—1942no—
356C22×C4—1942no—
357C22×C4C2×#8760411yescharacteristic, complemented
358C4⋊C4—971no—
359C4⋊C4—971no—
360C4⋊C4—971no—
361C4⋊C4—971no—
362C4⋊C4—971no—
363C4⋊C4—971no—
364C4⋊C4—3881no—
365C4⋊C4—1942no—
366C4⋊C4—1942no—
367C4⋊C4—1942no—
368C4⋊C4—1942no—
369C4⋊C4C2×#8760411yescharacteristic, complemented
370C4⋊C4C2×#8760411yescharacteristic, complemented
371C22⋊C4—974no—
372C22⋊C4—971no—
373C22⋊C4—971no—
374C22⋊C4—971no—
375C22⋊C4—971no—
376C22⋊C4—971no—
377C22⋊C4—971no—
378C22⋊C4—971no—
379C22⋊C4—971no—
380C22⋊C4—971no—
381C22⋊C4—3881no—
382C22⋊C4—1942no—
383C22⋊C4—1942no—
384C22⋊C4—1942no—
385C22⋊C4C2×#8760411yescharacteristic, complemented
386C22⋊C4C2×#8760411yescharacteristic, complemented
387C22⋊C4C2×#8760411yescharacteristic, complemented
388C42—3881no—
389C42—1942no—
390C42—1942no—
391C42—1942no—
392C42—1942no—
393C23—971no—
394C23—972no—
395C23—972no—
396C23—971no—
397C23—971no—
398C23—1941no—
399C23—1941no—
400C23—1941no—
401C23—1941no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23C22×#8760411yescharacteristic
409C23C22×#8760411yescharacteristic
410D4—22no—
411D4—1942no—
412D4—1942no—
413D4—1942no—
414C2×C4—22no—
415C2×C4—972no—
416C2×C4—972no—
417C2×C4—972no—
418C2×C4—972no—
419C2×C4—971no—
420C2×C4—971no—
421C2×C4—971no—
422C2×C4—971no—
423C2×C4—971no—
424C2×C4—3884no—
425C2×C4—3884no—
426C2×C4—3884no—
427C2×C4—1941no—
428C2×C4—1941no—
429C2×C4—1941no—
430C2×C4—1941no—
431C2×C4—1941no—
432C2×C4—1941no—
433C2×C4—1941no—
434C2×C4—1941no—
435C2×C4—1942no—
436C2×C4—1942no—
437C2×C4—1941no—
438C2×C4—1941no—
439C2×C4—1941no—
440C2×C4—1941no—
441C2×C4—1941no—
442C2×C4—1941no—
443C2×C4—1941no—
444C2×C4—1941no—
445C2×C4—1941no—
446C2×C4—1941no—
447C2×C4—1941no—
448C2×C4—1941no—
449C2×C4—1941no—
450C2×C4—1941no—
451C2×C4—1941no—
452C2×C4—1941no—
453C2×C4—1941no—
454C2×C4—1941no—
455C2×C4—1941no—
456C2×C4—1941no—
457C2×C4—1941no—
458C2×C4—1941no—
459C2×C4—1942no—
460C2×C4C22×#8760411yescharacteristic
461C2×C4C22×#8760411yescharacteristic
462C2×C4C22×#8760411yescharacteristic
463C2×C4C22×#8760411yescharacteristic
464C2×C4C22×#8760411yescharacteristic
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—971no—
471C22—971no—
472C22—971no—
473C22—971no—
474C22—971no—
475C22—971no—
476C22—972no—
477C22—3881no—
478C22—3881no—
479C22—3881no—
480C22—3881no—
481C22—1942no—
482C22—1942no—
483C22—1942no—
484C22—1942no—
485C22—1941no—
486C22—1941no—
487C22—1941no—
488C22—1941no—
489C22—1941no—
490C22D4×#8760412yes—
491C22C23×#8760411yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—3881no—
498C4—3881no—
499C4—3881no—
500C4—3881no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C4—1942no—
505C4—1941no—
506C4—1941no—
507C4—1941no—
508C4—1941no—
509C4—1941no—
510C2—22no—
511C2—41no—
512C2—971no—
513C2—971no—
514C2—971no—
515C2—971no—
516C2—3881no—
517C2—1942no—
518C2C2×D4×#8760411yescharacteristic
519C2#41×#8760411yescharacteristic
520C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#123×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenter#123#87604
hypocenterC97#2359
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C2×#15690889C2semidirect
3. #20261686C2semidirect
4. #20261685C2semidirect
5. #20261685C2semidirect
6. #20261684C2semidirect
7. #20261684C2semidirect
9. #20261682C2semidirect
10. #20261682C2semidirect
11. C4⋊C4×#87604C2semidirect
12. C22⋊C4×#87604C2semidirect
13. C22⋊C4×#87604C2semidirect
14. C2×(C2×#87604)⋊C4C2semidirect
48. #15690795C4semidirect
49. #15690795C4semidirect
50. #15690794C4semidirect
51. #15690793C4semidirect
52. #15690792C4semidirect
53. #15690792C4semidirect
54. #15690791C4semidirect
60. #123×C97C4semidirect
100. C22×Dic97C2×C4semidirect
103. C2×C4×D97C2×C4semidirect
104. C2×C4×D97C2×C4semidirect
110. C4⋊Dic97C2×C4semidirect
111. C4⋊Dic97C2×C4semidirect
112. #13095490C2×C4semidirect
113. #13095490C2×C4semidirect
114. #13095490C2×C4semidirect
115. #13095490C2×C4semidirect
116. C22⋊Dic97C2×C4semidirect
117. C22⋊Dic97C2×C4semidirect
118. C22⋊Dic97C2×C4semidirect
119. #13095488C2×C4semidirect
120. #13095488C2×C4semidirect
121. #13095488C2×C4semidirect
122. #13095488C2×C4semidirect
123. #13095488C2×C4semidirect
124. #13095488C2×C4semidirect
126. C22×C4×C97C2×C4semidirect
127. C4⋊C4×C97C2×C4semidirect
128. C4⋊C4×C97C2×C4semidirect
129. C22⋊C4×C97C2×C4semidirect
130. C22⋊C4×C97C2×C4semidirect
131. C22⋊C4×C97C2×C4semidirect
261. C97#2359semidirect
289. #123#87604semidirect
357. C22×C4C2×#87604semidirect
369. C4⋊C4C2×#87604semidirect
370. C4⋊C4C2×#87604semidirect
385. C22⋊C4C2×#87604semidirect
386. C22⋊C4C2×#87604semidirect
387. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#876042
C2×#156908892
#202616862
#202616852
#202616842
#202616832
#202616822
C4⋊C4×#876042
C22⋊C4×#876042
C2×(C2×#87604)⋊C42
#123×D972
#235997