Finite Group Databasebeta

← #27144579

Subgroups — #27144579

Order 12416.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690767C211yesmaximal, characteristic, complemented
2C22×#13095488C211yesmaximal, characteristic
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4#20260626C211yesmaximal, characteristic, complemented
5#20260626C211yesmaximal, characteristic, complemented
6#20260624C211yesmaximal, characteristic, complemented
7#20260624C211yesmaximal, characteristic, complemented
8#20260608C211yesmaximal, characteristic, complemented
9#20260607C211yesmaximal, characteristic, complemented
10#20260597C211yesmaximal, characteristic, complemented
11#20260594C211yesmaximal, characteristic, complemented
12#20260590C211yesmaximal, characteristic
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14C4×#13095488C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690767—24no—
21C2×#13095490C2211yescharacteristic, complemented
22C2×#13095490C2211yescharacteristic, complemented
23C2×#13095490C2211yescharacteristic, complemented
24C2×#13095490C2211yescharacteristic, complemented
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×#13095488—24no—
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C22⋊C4×D97C418yescomplemented
39C2×C4×Dic97C2211yescharacteristic, complemented
40C2×C4×Dic97C2211yescharacteristic, complemented
41#15690717C2211yescharacteristic, complemented
42#15690717C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47C23×C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C42×C97C2211yescharacteristic, complemented
51C2×C42×C97C2211yescharacteristic, complemented
52#95×C97C2211yescharacteristic, complemented
53#95×C97C2211yescharacteristic, complemented
54C23×D97—22no—
55C23×D97—22no—
56C23×D97—22no—
57C23×D97—22no—
58C23×D97—41no—
59C23×D97—41no—
60C23×D97C2×C414yes—
61C23×D97C2311yescharacteristic
62C23×D97C2311yescharacteristic
63C22×Dic97—22no—
64C22×Dic97—22no—
65C22×Dic97—41no—
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—41no—
75C2×C4×D97—41no—
76C2×C4×D97C2×C414yescomplemented
77C2×C4×D97C2×C414yescomplemented
78#13095490—22no—
79#13095490—22no—
80#13095490—41no—
81#13095490—41no—
82C22⋊Dic97—22no—
83C22⋊Dic97C2×C414yes—
84#13095488—22no—
85#13095488—22no—
86#13095488—22no—
87#13095488—22no—
88#13095488—22no—
89#13095488—41no—
90#13095488—41no—
91#13095488C2×C414yescomplemented
92#13095488C2×C414yescomplemented
93#13095488D414yes—
94C4×Dic97—22no—
95C4×Dic97—41no—
96C24×C97C2311yescharacteristic
97C22×C4×C97—22no—
98C22×C4×C97—22no—
99C22×C4×C97—41no—
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22⋊C4×C97—22no—
107C22⋊C4×C97C2×C414yes—
108C42×C97—22no—
109C42×C97—41no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97C22×C412yes—
115C23×C97#4112yes—
116C23×C97#4112yes—
117C23×C97C2411yescharacteristic
118C22×D97—21no—
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—28no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—28no—
130C22×D97—44no—
131C22×D97—44no—
132C22×D97—44no—
133C22×D97—44no—
134C22×D97—44no—
135C22×D97C22×C412yes—
136C22×D97C22×C412yes—
137C22×D97C2×D412yes—
138C22×D97#4112yes—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—21no—
153C2×C4×C97—21no—
154C2×C4×C97—44no—
155C2×C4×C97C22×C412yes—
156C2×C4×C97C22×C412yes—
157C2×C4×C97C2×D412yes—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—21no—
171C2×Dic97—21no—
172C2×Dic97—21no—
173C2×Dic97—44no—
174C2×Dic97C22×C412yes—
175C2×Dic97C22×C412yes—
176C2×Dic97C2×D412yes—
177C4×D97—28no—
178C4×D97—44no—
179C22×C97—22no—
180C22×C97—22no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97#11714yescomplemented
186C22×C97C23×C411yescharacteristic
187C22×C97C22×D411yescharacteristic
188C22×C97C2×#4111yescharacteristic
189C22×C97C2×#4111yescharacteristic
190C22×C97C2×#4111yescharacteristic
191C22×C97H(5,𝔽2)11yescharacteristic
192C22×C97H(5,𝔽2)11yescharacteristic
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—22no—
200C2×D97—22no—
201C2×D97—22no—
202C2×D97—416no—
203C2×D97—41no—
204C2×D97—41no—
205C2×D97—41no—
206C2×D97—41no—
207C2×D97C4×D414yes—
208C4×C97—22no—
209C4×C97—22no—
210C4×C97—22no—
211C4×C97—41no—
212C4×C97—41no—
213C4×C97—41no—
214Dic97—22no—
215Dic97—22no—
216Dic97—22no—
217Dic97—41no—
218Dic97—41no—
219Dic97—41no—
220C2×C97—24no—
221C2×C97C2×#11711yescharacteristic
222C2×C97C2×C4×D411yescharacteristic
223C2×C97#51811yescharacteristic
224C2×C97#54611yescharacteristic
225C2×C97#54611yescharacteristic
226C2×C97#55111yescharacteristic
227C2×C97#55111yescharacteristic
228D97—24no—
229D97—42no—
230#2342—971nomaximal, Sylow
231C97#234211yescharacteristic, complemented, Sylow
232C2×#117—971no—
233C22×C22⋊C4—971no—
234C22×C22⋊C4—971no—
235#388—971no—
236#388—971no—
237#388—971no—
238#388—971no—
239#385—971no—
240#385—971no—
241#381—971no—
242#380—971no—
243#379—971no—
244C4×C22⋊C4—971no—
245C4×C22⋊C4—971no—
246C4×C22⋊C4D9711yescharacteristic, complemented
247C25—971no—
248C23×C4—971no—
249C23×C4—971no—
250C23×C4—971no—
251C23×C4C2×D9711yescharacteristic
252#117—1944no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—971no—
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—978no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4—971no—
266C2×C22⋊C4—971no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—971no—
270C2×C22⋊C4—1944no—
271C2×C22⋊C4C2×D9711yescharacteristic
272C2×C22⋊C4C2×D9711yescharacteristic
273C2×C42—971no—
274C2×C42—971no—
275C2×C42C2×D9711yescharacteristic, complemented
276C2×C42C2×D9711yescharacteristic, complemented
277#95—971no—
278#95—971no—
279#95—971no—
280#95—971no—
281#95—971no—
282#95—971no—
283#95C2×D9711yescharacteristic, complemented
284#95C2×D9711yescharacteristic, complemented
285C24—971no—
286C24—974no—
287C24—971no—
288C24—3881no—
289C24—3881no—
290C24—1942no—
291C24—1942no—
292C24—1942no—
293C24—1942no—
294C24C22×D9711yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—971no—
299C22×C4—971no—
300C22×C4—971no—
301C22×C4—974no—
302C22×C4—974no—
303C22×C4—971no—
304C22×C4—971no—
305C22×C4—971no—
306C22×C4—3881no—
307C22×C4—3881no—
308C22×C4—3881no—
309C22×C4—1942no—
310C22×C4—1942no—
311C22×C4—1942no—
312C22×C4—1942no—
313C22×C4C22×D9711yescharacteristic
314C22×C4C22×D9711yescharacteristic
315C22×C4C22×D9711yescharacteristic
316C22×C4C22×D9711yescharacteristic
317C22×C4C22×D9711yescharacteristic
318C22×C4C22×D9711yescharacteristic
319C4⋊C4—3881no—
320C4⋊C4—3881no—
321C4⋊C4—1942no—
322C4⋊C4—1942no—
323C22⋊C4—22no—
324C22⋊C4—974no—
325C22⋊C4—974no—
326C22⋊C4—974no—
327C22⋊C4—974no—
328C22⋊C4—3881no—
329C22⋊C4—3881no—
330C22⋊C4—1942no—
331C22⋊C4—1942no—
332C22⋊C4—1942no—
333C22⋊C4—1942no—
334C22⋊C4—1942no—
335C22⋊C4—1942no—
336C22⋊C4C4×D9714yes—
337C42—22no—
338C42—41no—
339C42—3881no—
340C42—1942no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—972no—
346C23—972no—
347C23—972no—
348C23—972no—
349C23—3884no—
350C23—3884no—
351C23—3884no—
352C23—3884no—
353C23—3884no—
354C23—1948no—
355C23—1948no—
356C23—1941no—
357C23—1941no—
358C23—1941no—
359C23—1941no—
360C23—1941no—
361C23—1941no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23—1941no—
366C23C2×C4×D9712yes—
367C23#1309551112yes—
368C23#1309551112yes—
369C23C23×D9711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—972no—
387C2×C4—972no—
388C2×C4—972no—
389C2×C4—3884no—
390C2×C4—3884no—
391C2×C4—1948no—
392C2×C4—1941no—
393C2×C4—1941no—
394C2×C4—1941no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4—1941no—
402C2×C4—1941no—
403C2×C4—1941no—
404C2×C4—1941no—
405C2×C4—1941no—
406C2×C4—1941no—
407C2×C4C2×C4×D9712yes—
408C2×C4C2×C4×D9712yes—
409C2×C4D4×D9712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—974no—
417C22—38816no—
418C22—3881no—
419C22—3881no—
420C22—3881no—
421C22—3881no—
422C22—1942no—
423C22—1942no—
424C22—1942no—
425C22—1942no—
426C22—1942no—
427C22—1942no—
428C22—1942no—
429C22—1942no—
430C22—1942no—
431C22#1569076714yescomplemented
432C22C22×C4×D9711yescharacteristic
433C22C2×D4×D9711yescharacteristic
434C22#41×D9711yescharacteristic
435C22C2×#1309551111yescharacteristic
436C22C2×#1309551111yescharacteristic
437C22#1569085511yescharacteristic
438C22#1569085611yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—3881no—
446C4—3881no—
447C4—3881no—
448C4—1942no—
449C4—1942no—
450C4—1942no—
451C2—24no—
452C2—3882no—
453C2—1944no—
454C2C2×#1569076711yescharacteristic
455C2C4×D4×D9711yescharacteristic
456C2#2026120711yescharacteristic
457C2#2026137511yescharacteristic
458C2#2026138111yescharacteristic
459C2#2026140911yescharacteristic
460C2#2026141411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC4×C22⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenterC4×C22⋊C4D97
hypocenterC97#2342
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690767C2semidirect
4. #20260626C2semidirect
5. #20260626C2semidirect
6. #20260624C2semidirect
7. #20260624C2semidirect
8. #20260608C2semidirect
9. #20260607C2semidirect
10. #20260597C2semidirect
11. #20260594C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. C4×#13095488C2semidirect
15. C4×C22⋊C4×C97C2semidirect
21. C2×#13095490C22semidirect
22. C2×#13095490C22semidirect
23. C2×#13095490C22semidirect
24. C2×#13095490C22semidirect
38. C22⋊C4×D97C4semidirect
39. C2×C4×Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. #15690717C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
50. C2×C42×C97C22semidirect
51. C2×C42×C97C22semidirect
52. #95×C97C22semidirect
53. #95×C97C22semidirect
76. C2×C4×D97C2×C4semidirect
77. C2×C4×D97C2×C4semidirect
91. #13095488C2×C4semidirect
92. #13095488C2×C4semidirect
185. C22×C97#117semidirect
231. C97#2342semidirect
246. C4×C22⋊C4D97semidirect
275. C2×C42C2×D97semidirect
276. C2×C42C2×D97semidirect
283. #95C2×D97semidirect
284. #95C2×D97semidirect
431. C22#15690767semidirect

Maximal subgroups

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

subgroupindex
C2×#156907672
C22×#130954882
C2×C22⋊C4×D972
#202606262
#202606242
#202606082
#202606072
#202605972
#202605942
#202605902
C22⋊C4×Dic972
C4×#130954882
C4×C22⋊C4×C972
#234297