Finite Group Databasebeta

← #27158235

Subgroups — #27158235

Order 12416.

532 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261687C211yesmaximal, characteristic
2#20261686C211yesmaximal, characteristic, complemented
3#20261685C211yesmaximal, characteristic, complemented
4#20261685C211yesmaximal, characteristic, complemented
5#20261685C211yesmaximal, characteristic, complemented
6#20261685C211yesmaximal, characteristic, complemented
7#20261684C211yesmaximal, characteristic, complemented
8#20261684C211yesmaximal, characteristic, complemented
9#20261683C211yesmaximal, characteristic
10#20261682C211yesmaximal, characteristic, complemented
11#20261677C211yesmaximal, characteristic, complemented
12C4⋊C4×#87604C211yesmaximal, characteristic, complemented
13C22⋊C4×#87604C211yesmaximal, characteristic, complemented
14C4×C22⋊#87604C211yesmaximal, characteristic, complemented
15#123×D97C211yesmaximal, characteristic
16C23×#87604C2211yescharacteristic
17C2×C4⋊#87604C2211yescharacteristic
18C2×C4⋊#87604C2211yescharacteristic
19C2×C4⋊#87604C2211yescharacteristic
20C2×C4⋊#87604C2211yescharacteristic
21C2×C4⋊#87604C2211yescharacteristic
22C2×C22⋊#87604C2211yescharacteristic
23C2×C22⋊#87604C2211yescharacteristic
24C2×C22⋊#87604C2211yescharacteristic
25C2×C22⋊#87604C2211yescharacteristic
26C2×C22⋊#87604C2211yescharacteristic
27C2×C22⋊#87604C2211yescharacteristic
28C2×C22⋊#87604C2211yescharacteristic
29C2×C4×#87604C2211yescharacteristic
30C2×C4×#87604C2211yescharacteristic
31C2×C4×#87604C2211yescharacteristic
32C2×C4×#87604C2211yescharacteristic
33C2×C4×#87604C2211yescharacteristic
34(C2×#87604)⋊C4C2211yescharacteristic
35(C2×#87604)⋊C4C2211yescharacteristic
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
44C2×D4×D97C2211yescharacteristic
45C22×C4×D97C2211yescharacteristic
46#15690795C411yescharacteristic, complemented
47#15690795C411yescharacteristic, complemented
48#15690794C411yescharacteristic, complemented
49#15690793C411yescharacteristic, complemented
50#15690792C411yescharacteristic, complemented
51#15690792C411yescharacteristic, complemented
52#15690791C411yescharacteristic, complemented
53C4⋊C4×D97C2211yescharacteristic
54C4⋊C4×D97C2211yescharacteristic
55C22⋊C4×D97C2211yescharacteristic
56C22⋊C4×D97C2211yescharacteristic
57C22⋊C4×D97C2211yescharacteristic
58#123×C97C411yescharacteristic, complemented
59C22×#87604—22no—
60C22×#87604—41no—
61C22×#87604—41no—
62C22×#87604C2311yescharacteristic
63C22×#87604C2311yescharacteristic
64C22×#87604C2311yescharacteristic
65C22×#87604C2311yescharacteristic
66C22×#87604C2311yescharacteristic
67C22×#87604C2311yescharacteristic
68C22×#87604C2311yescharacteristic
69C22×#87604C2311yescharacteristic
70C4⋊#87604—22no—
71C4⋊#87604—41no—
72C4⋊#87604—41no—
73C4⋊#87604—41no—
74C4⋊#87604—41no—
75C22⋊#87604—22no—
76C22⋊#87604—22no—
77C22⋊#87604—22no—
78C22⋊#87604—41no—
79C22⋊#87604—41no—
80C22⋊#87604—41no—
81C22⋊#87604—41no—
82C4×#87604—22no—
83C4×#87604—41no—
84C4×#87604—41no—
85C4×#87604—41no—
86C4×#87604—41no—
87C23×D97C2311yescharacteristic
88C23×D97C2311yescharacteristic
89C2×D388C2×C411yescharacteristic
90C2×C22⋊D97C2×C411yescharacteristic
91C2×C22⋊D97C2×C411yescharacteristic
92D4×D97—41no—
93D4×D97—41no—
94C22×Dic97C2×C411yescharacteristic, complemented
95C2×C4×D97—41no—
96C2×C4×D97—41no—
97C2×C4×D97C2×C411yescharacteristic, complemented
98C2×C4×D97C2×C411yescharacteristic, complemented
99C2×C4×D97C2311yescharacteristic
100C2×C4×D97C2311yescharacteristic
101C2×C4×D97C2311yescharacteristic
102C2×C4×D97C2311yescharacteristic
103C2×C4×D97C2311yescharacteristic
104C4⋊Dic97C2×C411yescharacteristic, complemented
105C4⋊Dic97C2×C411yescharacteristic, complemented
106#13095490C2×C411yescharacteristic, complemented
107#13095490C2×C411yescharacteristic, complemented
108#13095490C2×C411yescharacteristic, complemented
109#13095490C2×C411yescharacteristic, complemented
110C22⋊Dic97C2×C411yescharacteristic
111C22⋊Dic97C2×C411yescharacteristic
112C22⋊Dic97C2×C411yescharacteristic, complemented
113#13095488C2×C411yescharacteristic
114#13095488C2×C411yescharacteristic
115#13095488C2×C411yescharacteristic
116#13095488C2×C411yescharacteristic
117#13095488C2×C411yescharacteristic, complemented
118#13095488C2×C411yescharacteristic, complemented
119C2×D4×C97C2×C411yescharacteristic
120C22×C4×C97C2×C411yescharacteristic, complemented
121C4⋊C4×C97C2×C411yescharacteristic, complemented
122C4⋊C4×C97C2×C411yescharacteristic, complemented
123C22⋊C4×C97C2×C411yescharacteristic
124C22⋊C4×C97C2×C411yescharacteristic
125C22⋊C4×C97C2×C411yescharacteristic, complemented
126C23×C97C22×C411yescharacteristic
127C23×C97C22×C411yescharacteristic
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—21no—
134C22×D97—21no—
135C22×D97—21no—
136C22×D97—21no—
137C22×D97—21no—
138C22×D97—21no—
139C22×D97—21no—
140C22×D97C22×C411yescharacteristic
141C22×D97C22×C411yescharacteristic
142C22×D97C2411yescharacteristic
143C2×#87604—21no—
144C2×#87604—21no—
145C2×#87604—21no—
146C2×#87604—21no—
147C2×#87604—21no—
148C2×#87604—21no—
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—44no—
166C2×#87604#4112yes—
167C2×#87604#4112yes—
168D4×C97—41no—
169C2×C4×C97—41no—
170C2×C4×C97C22×C411yescharacteristic
171C2×C4×C97C22×C411yescharacteristic
172C2×C4×C97C22×C411yescharacteristic
173C2×C4×C97C22×C411yescharacteristic
174C2×C4×C97C22×C411yescharacteristic
175C22⋊D97—41no—
176C22⋊D97—41no—
177C2×Dic97—41no—
178C2×Dic97C22×C411yescharacteristic
179C2×Dic97C22×C411yescharacteristic
180C2×Dic97C22×C411yescharacteristic
181C2×Dic97C22×C411yescharacteristic
182C2×Dic97C22×C411yescharacteristic
183D388—41no—
184C4×D97—21no—
185C4×D97—21no—
186C4×D97—21no—
187C4×D97—21no—
188C4×D97—21no—
189C4×D97—21no—
190C4×D97—21no—
191C4×D97—21no—
192C4×D97—21no—
193C4×D97—21no—
194C4×D97—41no—
195C4×D97—41no—
196C22×C97—21no—
197C22×C97—21no—
198C22×C97—21no—
199C22×C97—21no—
200C22×C97—21no—
201C22×C97—21no—
202C22×C97C23×C411yescharacteristic
203C2×D97—21no—
204C2×D97—21no—
205C2×D97—21no—
206C2×D97—21no—
207C2×D97—21no—
208C2×D97—21no—
209C2×D97—41no—
210C2×D97—41no—
211C2×D97—41no—
212C2×D97—41no—
213C2×D97—41no—
214C2×D97—41no—
215C2×D97—41no—
216C2×D97—41no—
217C2×D97C2×#4111yescharacteristic
218C2×D97C2×#4111yescharacteristic
219C2×D97H(5,𝔽2)11yescharacteristic
220C2×D97H(5,𝔽2)11yescharacteristic
221C2×D97H(5,𝔽2)11yescharacteristic
222C2×D972−1+411yescharacteristic
223#87604—22no—
224#87604—22no—
225#87604—41no—
226#87604—41no—
227#87604—41no—
228#87604—41no—
229#87604—41no—
230#87604—41no—
231C4×C97—21no—
232C4×C97—21no—
233C4×C97—21no—
234C4×C97—21no—
235C4×C97—21no—
236Dic97—21no—
237Dic97—21no—
238Dic97—21no—
239Dic97—21no—
240Dic97—21no—
241C2×C97—41no—
242C2×C97—41no—
243C2×C97C4×#4111yescharacteristic
244C2×C97#51811yescharacteristic
245C2×C97#52011yescharacteristic
246D97—41no—
247D97—41no—
248D97#53811yescharacteristic
249D97#53911yescharacteristic
250D97#54011yescharacteristic
251D97#54211yescharacteristic
252#2335—971nomaximal, Sylow
253C97#233511yescharacteristic, complemented, Sylow
254C2×#123—971no—
255#390—971no—
256#389—971no—
257#388—971no—
258#388—971no—
259#388—971no—
260#388—971no—
261#387—971no—
262#387—971no—
263#386—971no—
264#385—971no—
265#380—971no—
266C4×C4⋊C4—971no—
267C4×C22⋊C4—971no—
268C4×C22⋊C4—971no—
269C22×D4—971no—
270C23×C4—971no—
271C23×C4—971no—
272#123—971no—
273#123—971no—
274#123—971no—
275#123—971no—
276#123—971no—
277#123—971no—
278#123—971no—
279#123#8760411yescharacteristic, complemented
280C2×C4⋊C4—971no—
281C2×C4⋊C4—971no—
282C2×C4⋊C4—971no—
283C2×C4⋊C4—971no—
284C2×C4⋊C4—971no—
285C2×C4⋊C4—971no—
286C2×C4⋊C4—971no—
287C2×C22⋊C4—971no—
288C2×C22⋊C4—971no—
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—971no—
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4—971no—
295C2×C22⋊C4—971no—
296C2×C22⋊C4—971no—
297C2×C42—971no—
298C2×C42—971no—
299C2×C42—971no—
300C2×C42—971no—
301C2×C42—971no—
302#95—971no—
303#95—971no—
304#95—971no—
305#95—971no—
306#95—971no—
307#95—971no—
308#95—971no—
309#95—971no—
310#95—971no—
311#95—971no—
312C24—971no—
313C24—971no—
314C2×D4—971no—
315C2×D4—971no—
316C2×D4—971no—
317C2×D4—3881no—
318C2×D4—3881no—
319C2×D4C2×#8760411yescharacteristic
320C22×C4—971no—
321C22×C4—971no—
322C22×C4—971no—
323C22×C4—971no—
324C22×C4—971no—
325C22×C4—971no—
326C22×C4—971no—
327C22×C4—971no—
328C22×C4—971no—
329C22×C4—971no—
330C22×C4—971no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—971no—
334C22×C4—971no—
335C22×C4—971no—
336C22×C4—3881no—
337C22×C4—3881no—
338C22×C4—3881no—
339C22×C4—3881no—
340C22×C4—1942no—
341C22×C4C2×#8760411yescharacteristic, complemented
342C4⋊C4—971no—
343C4⋊C4—971no—
344C4⋊C4—971no—
345C4⋊C4—971no—
346C4⋊C4—971no—
347C4⋊C4—971no—
348C4⋊C4—3881no—
349C4⋊C4—3881no—
350C4⋊C4—3881no—
351C4⋊C4—3881no—
352C4⋊C4—1942no—
353C4⋊C4C2×#8760411yescharacteristic, complemented
354C4⋊C4C2×#8760411yescharacteristic, complemented
355C22⋊C4—971no—
356C22⋊C4—971no—
357C22⋊C4—971no—
358C22⋊C4—971no—
359C22⋊C4—971no—
360C22⋊C4—971no—
361C22⋊C4—971no—
362C22⋊C4—971no—
363C22⋊C4—971no—
364C22⋊C4—3881no—
365C22⋊C4—3881no—
366C22⋊C4—3881no—
367C22⋊C4—3881no—
368C22⋊C4—1942no—
369C22⋊C4—1942no—
370C22⋊C4—1942no—
371C22⋊C4C2×#8760411yescharacteristic
372C22⋊C4C2×#8760411yescharacteristic
373C22⋊C4C2×#8760411yescharacteristic, complemented
374C42—3881no—
375C42—3881no—
376C42—3881no—
377C42—3881no—
378C42—1942no—
379C23—971no—
380C23—971no—
381C23—971no—
382C23—1941no—
383C23—1941no—
384C23—1941no—
385C23—1941no—
386C23—1941no—
387C23—1941no—
388C23—1941no—
389C23—1941no—
390C23—1941no—
391C23—1941no—
392C23—1941no—
393C23—1941no—
394C23C22×#8760411yescharacteristic
395C23C22×#8760411yescharacteristic
396D4—41no—
397D4—3881no—
398D4—3881no—
399D4—3881no—
400C2×C4—41no—
401C2×C4—972no—
402C2×C4—972no—
403C2×C4—971no—
404C2×C4—971no—
405C2×C4—971no—
406C2×C4—971no—
407C2×C4—971no—
408C2×C4—3881no—
409C2×C4—3881no—
410C2×C4—3884no—
411C2×C4—3881no—
412C2×C4—1941no—
413C2×C4—1941no—
414C2×C4—1941no—
415C2×C4—1941no—
416C2×C4—1941no—
417C2×C4—1941no—
418C2×C4—1941no—
419C2×C4—1941no—
420C2×C4—1941no—
421C2×C4—1941no—
422C2×C4—1941no—
423C2×C4—1941no—
424C2×C4—1941no—
425C2×C4—1941no—
426C2×C4—1941no—
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—1941no—
436C2×C4—1941no—
437C2×C4—1941no—
438C2×C4—1941no—
439C2×C4—1941no—
440C2×C4—1941no—
441C2×C4—1941no—
442C2×C4—1941no—
443C2×C4—1941no—
444C2×C4C22×#8760411yescharacteristic
445C2×C4C22×#8760411yescharacteristic
446C2×C4C22×#8760411yescharacteristic
447C2×C4C22×#8760411yescharacteristic
448C2×C4C22×#8760411yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—971no—
456C22—971no—
457C22—971no—
458C22—971no—
459C22—971no—
460C22—971no—
461C22—3881no—
462C22—3881no—
463C22—3881no—
464C22—3881no—
465C22—3881no—
466C22—3881no—
467C22—3881no—
468C22—3881no—
469C22—1941no—
470C22—1941no—
471C22—1941no—
472C22—1941no—
473C22—1941no—
474C22—1941no—
475C22C23×#8760411yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—3881no—
482C4—3881no—
483C4—3881no—
484C4—3881no—
485C4—3881no—
486C4—3881no—
487C4—1942no—
488C4—1942no—
489C4—1941no—
490C4—1941no—
491C4—1941no—
492C4—1941no—
493C4—1941no—
494C2—41no—
495C2—41no—
496C2—971no—
497C2—971no—
498C2—971no—
499C2—971no—
500C2—3881no—
501C2—3881no—
502C2#41×#8760411yescharacteristic
503C2#2026180211yescharacteristic
504C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#123×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenter#123#87604
hypocenterC97#2335
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20261686C2semidirect
3. #20261685C2semidirect
4. #20261685C2semidirect
5. #20261685C2semidirect
6. #20261685C2semidirect
7. #20261684C2semidirect
8. #20261684C2semidirect
10. #20261682C2semidirect
11. #20261677C2semidirect
12. C4⋊C4×#87604C2semidirect
13. C22⋊C4×#87604C2semidirect
14. C4×C22⋊#87604C2semidirect
46. #15690795C4semidirect
47. #15690795C4semidirect
48. #15690794C4semidirect
49. #15690793C4semidirect
50. #15690792C4semidirect
51. #15690792C4semidirect
52. #15690791C4semidirect
58. #123×C97C4semidirect
94. C22×Dic97C2×C4semidirect
97. C2×C4×D97C2×C4semidirect
98. C2×C4×D97C2×C4semidirect
104. C4⋊Dic97C2×C4semidirect
105. C4⋊Dic97C2×C4semidirect
106. #13095490C2×C4semidirect
107. #13095490C2×C4semidirect
108. #13095490C2×C4semidirect
109. #13095490C2×C4semidirect
112. C22⋊Dic97C2×C4semidirect
117. #13095488C2×C4semidirect
118. #13095488C2×C4semidirect
120. C22×C4×C97C2×C4semidirect
121. C4⋊C4×C97C2×C4semidirect
122. C4⋊C4×C97C2×C4semidirect
125. C22⋊C4×C97C2×C4semidirect
253. C97#2335semidirect
279. #123#87604semidirect
341. C22×C4C2×#87604semidirect
353. C4⋊C4C2×#87604semidirect
354. C4⋊C4C2×#87604semidirect
373. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202616872
#202616862
#202616852
#202616842
#202616832
#202616822
#202616772
C4⋊C4×#876042
C22⋊C4×#876042
C4×C22⋊#876042
#123×D972
#233597