Finite Group Databasebeta

← #388×D97 #27144164

Subgroups — #388×D97 #27144164

Order 12416.

928 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D97C211yesmaximal, characteristic
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4C2×C22⋊C4×D97C211yesmaximal, characteristic
5C2×C42×D97C211yesmaximal, characteristic, complemented
6#20260629C211yesmaximal, characteristic, complemented
7#20260628C211yesmaximal, characteristic, complemented
8#20260627C211yesmaximal, characteristic, complemented
9#20260626C211yesmaximal, characteristic, complemented
10#20260625C211yesmaximal, characteristic, complemented
11#20260624C211yesmaximal, characteristic, complemented
12#20260623C211yesmaximal, characteristic, complemented
13#95×D97C211yesmaximal, characteristic, complemented
14#95×D97C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C2×C4⋊Dic97C2211yescharacteristic, complemented
24C2×#13095490C2211yescharacteristic, complemented
25C2×#13095490C2211yescharacteristic, complemented
26C4⋊C4×D97—24no—
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C22⋊C4×D97—24no—
37C22⋊C4×D97—24no—
38C22⋊C4×D97C418yescomplemented
39C2×C4×Dic97C2211yescharacteristic, complemented
40C2×C4×Dic97C2211yescharacteristic, complemented
41C2×C4×Dic97C2211yescharacteristic, complemented
42C42×D97—24no—
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47#15690717C2211yescharacteristic, complemented
48#15690717C2211yescharacteristic, complemented
49C2×C4⋊C4×C97C2211yescharacteristic, complemented
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C42×C97C2211yescharacteristic, complemented
54#95×C97C2211yescharacteristic, complemented
55#95×C97C2211yescharacteristic, complemented
56C23×D97—22no—
57C23×D97—22no—
58C23×D97—22no—
59C23×D97—22no—
60C23×D97—22no—
61C23×D97—22no—
62C23×D97C2×C414yes—
63C23×D97C2311yescharacteristic
64C23×D97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—22no—
78C2×C4×D97—22no—
79C2×C4×D97—22no—
80C2×C4×D97—22no—
81C2×C4×D97—22no—
82C2×C4×D97—22no—
83C2×C4×D97—22no—
84C2×C4×D97—22no—
85C2×C4×D97C2×C414yes—
86C2×C4×D97C2×C414yes—
87C2×C4×D97D414yes—
88C2×C4×D97D414yes—
89C4⋊Dic97—22no—
90#13095490—22no—
91#13095490—22no—
92C22⋊Dic97—22no—
93C22⋊Dic97—22no—
94C22⋊Dic97C2×C414yescomplemented
95#13095488—22no—
96#13095488—22no—
97#13095488—22no—
98#13095488—22no—
99#13095488C2×C414yescomplemented
100#13095488C2×C414yescomplemented
101C4×Dic97—22no—
102C4×Dic97—22no—
103C4×Dic97—22no—
104C24×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C22×C4×C97C2311yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C4⋊C4×C97—22no—
112C22⋊C4×C97—22no—
113C22⋊C4×C97—22no—
114C22⋊C4×C97C2×C414yescomplemented
115C42×C97—22no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C23×C97—21no—
122C23×C97C22×C412yes—
123C23×C97C2411yescharacteristic
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—28no—
131C22×D97—28no—
132C22×D97—28no—
133C22×D97—44no—
134C22×D97—44no—
135C22×D97—44no—
136C22×D97—44no—
137C22×D97C22×C412yes—
138C22×D97C22×C412yes—
139C22×D97C2×D412yes—
140C22×D97C2×D412yes—
141C22×D97#4112yes—
142C22×D97#4112yes—
143C22×D97#4112yes—
144C22×D97#4112yes—
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—21no—
155C2×C4×C97—21no—
156C2×C4×C97—21no—
157C2×C4×C97—21no—
158C2×C4×C97—21no—
159C2×C4×C97C22×C412yes—
160C2×C4×C97C22×C412yes—
161C2×C4×C97C2×D412yes—
162C2×C4×C97C2×D412yes—
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—21no—
174C2×Dic97—21no—
175C2×Dic97—21no—
176C2×Dic97—21no—
177C2×Dic97C22×C412yes—
178C2×Dic97C22×C412yes—
179C2×Dic97C2×D412yes—
180C2×Dic97C2×D412yes—
181C4×D97—28no—
182C4×D97—28no—
183C4×D97—28no—
184C4×D97—28no—
185C4×D97—44no—
186C4×D97—44no—
187C22×C97—22no—
188C22×C97—22no—
189C22×C97—22no—
190C22×C97—41no—
191C22×C97—41no—
192C22×C97—41no—
193C22×C97—41no—
194C22×C97C23×C411yescharacteristic
195C22×C97C22×D411yescharacteristic
196C22×C97C22×D411yescharacteristic
197C22×C97C2×#4111yescharacteristic
198C22×C97C2×#4111yescharacteristic
199C22×C97C2×#4111yescharacteristic
200C22×C97C2×#4111yescharacteristic
201C2×D97—22no—
202C2×D97—22no—
203C2×D97—22no—
204C2×D97—41no—
205C2×D97—41no—
206C2×D97—41no—
207C2×D97—41no—
208C2×D97—416no—
209C2×D97#11714yes—
210C2×D97C4×D414yes—
211C2×D97C4×D414yes—
212C2×D97#12114yes—
213C2×D97#12314yes—
214C2×D97#12414yes—
215C2×D97#12614yes—
216C4×C97—22no—
217C4×C97—22no—
218C4×C97—22no—
219C4×C97—22no—
220C4×C97—41no—
221C4×C97—41no—
222Dic97—22no—
223Dic97—22no—
224Dic97—22no—
225Dic97—22no—
226Dic97—41no—
227Dic97—41no—
228C2×C97—42no—
229C2×C97C2×#11711yescharacteristic
230C2×C97C2×C4×D411yescharacteristic
231C2×C97C2×C4×D411yescharacteristic
232C2×C97C2×#12111yescharacteristic
233C2×C97C2×#12311yescharacteristic
234C2×C97C2×#12411yescharacteristic
235C2×C97C2×#12611yescharacteristic
236D97—42no—
237D97#38818yesdirect product factor
238C2×#388—971nomaximal, Sylow
239C97C2×#38811yescharacteristic, complemented, Sylow
240C22×C4⋊C4—971no—
241C22×C22⋊C4—971no—
242C22×C22⋊C4—971no—
243C22×C22⋊C4—971no—
244C22×C42—971no—
245#388—971no—
246#388—971no—
247#388—971no—
248#388—971no—
249#388—971no—
250#388—971no—
251#388—971no—
252#388D9711yescharacteristic, direct product factor
253C2×#95—971no—
254C2×#95—971no—
255C25—971no—
256C23×C4—971no—
257C23×C4—971no—
258C23×C4—971no—
259C23×C4—971no—
260C23×C4—971no—
261C23×C4—971no—
262C2×C4⋊C4—971no—
263C2×C4⋊C4—971no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—1944no—
266C2×C4⋊C4C2×D9711yescharacteristic, complemented
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—971no—
270C2×C22⋊C4—978no—
271C2×C22⋊C4—971no—
272C2×C22⋊C4—971no—
273C2×C22⋊C4—971no—
274C2×C22⋊C4—971no—
275C2×C22⋊C4—971no—
276C2×C22⋊C4—971no—
277C2×C22⋊C4—1944no—
278C2×C22⋊C4—1944no—
279C2×C22⋊C4C2×D9711yescharacteristic
280C2×C22⋊C4C2×D9711yescharacteristic
281C2×C22⋊C4C2×D9711yescharacteristic
282C2×C42—971no—
283C2×C42—971no—
284C2×C42—971no—
285C2×C42—1944no—
286C2×C42C2×D9711yescharacteristic, complemented
287#95—971no—
288#95—971no—
289#95—971no—
290#95—971no—
291#95—971no—
292#95—971no—
293#95C2×D9711yescharacteristic, complemented
294#95C2×D9711yescharacteristic, complemented
295C24—971no—
296C24—974no—
297C24—971no—
298C24—1942no—
299C24—1942no—
300C24—1942no—
301C24—1942no—
302C24—1942no—
303C24—1942no—
304C24C22×D9711yescharacteristic
305C22×C4—971no—
306C22×C4—971no—
307C22×C4—971no—
308C22×C4—971no—
309C22×C4—974no—
310C22×C4—974no—
311C22×C4—974no—
312C22×C4—974no—
313C22×C4—971no—
314C22×C4—971no—
315C22×C4—1942no—
316C22×C4—1942no—
317C22×C4—1942no—
318C22×C4—1942no—
319C22×C4—1942no—
320C22×C4—1942no—
321C22×C4—1942no—
322C22×C4—1942no—
323C22×C4—1942no—
324C22×C4—1942no—
325C22×C4—1942no—
326C22×C4—1942no—
327C22×C4—1942no—
328C22×C4—1942no—
329C22×C4C22×D9711yescharacteristic
330C22×C4C22×D9711yescharacteristic
331C22×C4C22×D9711yescharacteristic
332C22×C4C22×D9711yescharacteristic
333C22×C4C22×D9711yescharacteristic
334C22×C4C22×D9711yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—1942no—
337C4⋊C4—1942no—
338C4⋊C4—1942no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—974no—
342C22⋊C4—974no—
343C22⋊C4—974no—
344C22⋊C4—1942no—
345C22⋊C4—1942no—
346C22⋊C4—1942no—
347C22⋊C4—1942no—
348C22⋊C4—1942no—
349C22⋊C4—1942no—
350C22⋊C4C4×D9714yescomplemented
351C42—22no—
352C42—1942no—
353C42—1942no—
354C42—1942no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—972no—
362C23—972no—
363C23—972no—
364C23—972no—
365C23—972no—
366C23—972no—
367C23—972no—
368C23—972no—
369C23—3884no—
370C23—3884no—
371C23—3884no—
372C23—3884no—
373C23—1948no—
374C23—1948no—
375C23—1948no—
376C23—1941no—
377C23—1941no—
378C23—1941no—
379C23—1941no—
380C23—1941no—
381C23—1941no—
382C23C2×C4×D9712yes—
383C23C23×D9711yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—972no—
399C2×C4—972no—
400C2×C4—972no—
401C2×C4—972no—
402C2×C4—3884no—
403C2×C4—3884no—
404C2×C4—1948no—
405C2×C4—1948no—
406C2×C4—1948no—
407C2×C4—1948no—
408C2×C4—1941no—
409C2×C4—1941no—
410C2×C4—1941no—
411C2×C4—1941no—
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×C4C2×C4×D9712yes—
423C2×C4C2×C4×D9712yes—
424C2×C4D4×D9712yes—
425C2×C4D4×D9712yes—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—974no—
434C22—974no—
435C22—974no—
436C22—974no—
437C22—974no—
438C22—974no—
439C22—974no—
440C22—38816no—
441C22—3881no—
442C22—3881no—
443C22—3881no—
444C22—3881no—
445C22—1942no—
446C22—1942no—
447C22—1942no—
448C22C22×C4×D9711yescharacteristic
449C22C2×D4×D9711yescharacteristic
450C22C2×D4×D9711yescharacteristic
451C22#41×D9711yescharacteristic
452C22#41×D9711yescharacteristic
453C22#41×D9711yescharacteristic
454C22#41×D9711yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—3881no—
462C4—3881no—
463C4—1942no—
464C4—1942no—
465C4—1942no—
466C4—1942no—
467C2—42no—
468C2—978no—
469C2—3882no—
470C2#117×D9711yescharacteristic
471C2C4×D4×D9711yescharacteristic
472C2C4×D4×D9711yescharacteristic
473C2#121×D9711yescharacteristic
474C2#123×D9711yescharacteristic
475C2#124×D9711yescharacteristic
476C2#126×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#388D97
hypocenterC97C2×#388
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D97C2semidirect
5. C2×C42×D97C2semidirect
6. #20260629C2semidirect
7. #20260628C2semidirect
8. #20260627C2semidirect
9. #20260626C2semidirect
10. #20260625C2semidirect
11. #20260624C2semidirect
12. #20260623C2semidirect
13. #95×D97C2semidirect
14. #95×D97C2semidirect
15. #388×C97C2semidirect
23. C2×C4⋊Dic97C22semidirect
24. C2×#13095490C22semidirect
25. C2×#13095490C22semidirect
38. C22⋊C4×D97C4semidirect
39. C2×C4×Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. C2×C4×Dic97C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
48. #15690717C22semidirect
49. C2×C4⋊C4×C97C22semidirect
53. C2×C42×C97C22semidirect
54. #95×C97C22semidirect
55. #95×C97C22semidirect
94. C22⋊Dic97C2×C4semidirect
99. #13095488C2×C4semidirect
100. #13095488C2×C4semidirect
114. C22⋊C4×C97C2×C4semidirect
237. D97#388direct (N × H)
239. C97C2×#388semidirect
252. #388D97direct (N × H)
266. C2×C4⋊C4C2×D97semidirect
286. C2×C42C2×D97semidirect
293. #95C2×D97semidirect
294. #95C2×D97semidirect
350. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C2×C22⋊C4×D972
C2×C42×D972
#202606292
#202606282
#202606272
#202606262
#202606252
#202606242
#202606232
#95×D972
#388×C972
C2×#38897