Finite Group Databasebeta

← #27144246

Subgroups — #27144246

Order 12416.

824 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690782C211yesmaximal, characteristic
2C2×D4×Dic97C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4C2×#15690770C211yesmaximal, characteristic, complemented
5C22×C4×Dic97C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260632C211yesmaximal, characteristic, complemented
8#20260627C211yesmaximal, characteristic, complemented
9#20260623C211yesmaximal, characteristic, complemented
10#20260614C211yesmaximal, characteristic, complemented
11#20260593C211yesmaximal, characteristic, complemented
12#20260586C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#386×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22#15690782C418yescomplemented
23D4×Dic97—24no—
24#15690770—24no—
25#15690770—24no—
26C2×C4⋊Dic97C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×#13095490C2211yescharacteristic
30C2×#13095490C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×C22⋊Dic97C2211yescharacteristic
33C2×C22⋊Dic97C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×C4×Dic97—24no—
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41C2×C4×Dic97C2211yescharacteristic
42C2×C4×Dic97C2211yescharacteristic
43C2×C4×Dic97C2211yescharacteristic
44C2×C4×Dic97C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47#15690717C2211yescharacteristic
48#15690717C2211yescharacteristic
49C22×D4×C97C2211yescharacteristic
50C23×C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C22⋊C4×C97C2211yescharacteristic
54#95×C97C2211yescharacteristic
55#95×C97C2211yescharacteristic
56C23×D97C2311yescharacteristic
57C2×C22⋊D97—22no—
58C2×C22⋊D97—22no—
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97C2×C414yescomplemented
62C2×C22⋊D97C2×C414yescomplemented
63C22×Dic97—22no—
64C22×Dic97—22no—
65C22×Dic97—22no—
66C22×Dic97—22no—
67C22×Dic97—22no—
68C22×Dic97—22no—
69C22×Dic97—22no—
70C22×Dic97C2×C414yescomplemented
71C22×Dic97D414yescomplemented
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C2×C4×D97—22no—
80C2×C4×D97—22no—
81C2×C4×D97—22no—
82C4⋊Dic97—22no—
83#13095490—22no—
84#13095490—22no—
85#13095490—22no—
86#13095490C2×C414yescomplemented
87C22⋊Dic97—22no—
88C22⋊Dic97—22no—
89C22⋊Dic97C2×C414yescomplemented
90#13095488—22no—
91#13095488—22no—
92#13095488—22no—
93#13095488C2×C414yescomplemented
94C4×Dic97—22no—
95C4×Dic97—22no—
96C4×Dic97—22no—
97C4×Dic97—22no—
98C4×Dic97—22no—
99C4×Dic97D414yescomplemented
100C24×C97C2311yescharacteristic
101C24×C97C2311yescharacteristic
102C2×D4×C97—22no—
103C2×D4×C97—22no—
104C2×D4×C97C2×C414yescomplemented
105C22×C4×C97—22no—
106C22×C4×C97—22no—
107C22×C4×C97—22no—
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22⋊C4×C97—22no—
114C22⋊C4×C97—22no—
115C22⋊C4×C97—22no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C23×C97—21no—
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97C22×C412yes—
127C23×C97C22×C412yes—
128C23×C97C2×D412yes—
129C23×C97#4112yes—
130C23×C97C2411yescharacteristic
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—21no—
134C22×D97—21no—
135C22×D97—21no—
136C22×D97—21no—
137C22×D97C22×C412yes—
138D4×C97—28no—
139C2×C4×C97—28no—
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×C97C22×C412yes—
153C2×C4×C97C2×D412yes—
154C2×C4×C97#4112yes—
155C22⋊D97—28no—
156C22⋊D97—28no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—28no—
164C2×Dic97—28no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—21no—
171C2×Dic97—44no—
172C2×Dic97C22×C412yes—
173C2×Dic97C22×C412yes—
174C2×Dic97C22×C412yes—
175C2×Dic97C2×D412yes—
176C2×Dic97C2×D412yes—
177C2×Dic97C2×D412yes—
178C2×Dic97C2×D412yes—
179C2×Dic97#4112yes—
180C2×Dic97#4112yes—
181C4×D97—44no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—22no—
186C22×C97—22no—
187C22×C97—22no—
188C22×C97—22no—
189C22×C97—22no—
190C22×C97—22no—
191C22×C97—41no—
192C22×C97—41no—
193C22×C97—41no—
194C22×C97—41no—
195C22×C97C4×D414yescomplemented
196C22×C97C23×C411yescharacteristic
197C22×C97C22×D411yescharacteristic
198C22×C97C22×D411yescharacteristic
199C22×C97C2×#4111yescharacteristic
200C22×C97C2×#4111yescharacteristic
201C22×C97C2×#4111yescharacteristic
202C22×C97C2×#4111yescharacteristic
203C2×D97—22no—
204C2×D97—22no—
205C2×D97—22no—
206C2×D97—41no—
207C2×D97—41no—
208C2×D97—41no—
209C2×D97—41no—
210C4×C97—22no—
211C4×C97—22no—
212C4×C97—22no—
213C4×C97—41no—
214C4×C97—41no—
215Dic97—22no—
216Dic97—22no—
217Dic97—22no—
218Dic97—41no—
219Dic97—41no—
220Dic97C4×D414yescomplemented
221Dic97#12114yescomplemented
222C2×C97—24no—
223C2×C97—42no—
224C2×C97C2×C4×D411yescharacteristic
225C2×C97C2×C4×D411yescharacteristic
226C2×C97C4×#4111yescharacteristic
227C2×C97C2×#12111yescharacteristic
228C2×C97#52511yescharacteristic
229C2×C97#52911yescharacteristic
230C2×C97#53211yescharacteristic
231D97—42no—
232C4×#121—971nomaximal, Sylow
233C97C4×#12111yescharacteristic, complemented, Sylow
234C2×#121—971no—
235C2×C4×D4—971no—
236C2×C4×D4—971no—
237C2×C4×D4—971no—
238C22×C42—971no—
239#390—971no—
240#389—971no—
241#388—971no—
242#388—971no—
243#386—971no—
244#386D9711yescharacteristic, complemented
245#380—971no—
246C4×C4⋊C4—971no—
247C4×C22⋊C4—971no—
248C4×C22⋊C4—971no—
249C22×D4—971no—
250C22×D4—971no—
251C22×D4C2×D9711yescharacteristic
252C23×C4—971no—
253C23×C4—971no—
254C23×C4—971no—
255C23×C4—971no—
256C23×C4C2×D9711yescharacteristic
257#121—978no—
258C4×D4—1944no—
259C4×D4—1944no—
260C4×D4—1944no—
261C2×C4⋊C4—971no—
262C2×C4⋊C4—971no—
263C2×C4⋊C4—971no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—971no—
266C2×C22⋊C4—971no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—971no—
270C2×C22⋊C4—971no—
271C2×C22⋊C4—971no—
272C2×C22⋊C4—971no—
273C2×C22⋊C4C2×D9711yescharacteristic
274C2×C22⋊C4C2×D9711yescharacteristic
275C2×C22⋊C4C2×D9711yescharacteristic
276C2×C42—971no—
277C2×C42—971no—
278C2×C42—971no—
279C2×C42—971no—
280C2×C42—971no—
281C2×C42—971no—
282C2×C42—1944no—
283#95—971no—
284#95—971no—
285#95—971no—
286#95—971no—
287#95C2×D9711yescharacteristic
288#95C2×D9711yescharacteristic
289C24—971no—
290C24C22×D9711yescharacteristic
291C24C22×D9711yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—974no—
295C2×D4—974no—
296C2×D4—1942no—
297C2×D4—1942no—
298C2×D4—1942no—
299C2×D4—1942no—
300C2×D4C4×D9714yescomplemented
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—971no—
305C22×C4—971no—
306C22×C4—974no—
307C22×C4—974no—
308C22×C4—971no—
309C22×C4—971no—
310C22×C4—971no—
311C22×C4—971no—
312C22×C4—971no—
313C22×C4—1942no—
314C22×C4—1942no—
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×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9711yescharacteristic
326C22×C4C22×D9711yescharacteristic
327C22×C4C22×D9711yescharacteristic
328C4⋊C4—974no—
329C4⋊C4—1942no—
330C4⋊C4—1942no—
331C4⋊C4—1942no—
332C4⋊C4—1942no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—974no—
337C22⋊C4—974no—
338C22⋊C4—1942no—
339C22⋊C4—1942no—
340C22⋊C4—1942no—
341C22⋊C4—1942no—
342C22⋊C4—1942no—
343C42—974no—
344C42—1942no—
345C42—1942no—
346C42—1942no—
347C42—1942no—
348C42—1942no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—972no—
360C23—1941no—
361C23—1941no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23—1941no—
366C23C2×C4×D9712yes—
367C23C2×C4×D9712yes—
368C23D4×D9712yes—
369C23#1309551012yes—
370C23C23×D9711yescharacteristic
371D4—28no—
372D4—1948no—
373D4—1948no—
374C2×C4—28no—
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—21no—
386C2×C4—21no—
387C2×C4—972no—
388C2×C4—972no—
389C2×C4—972no—
390C2×C4—972no—
391C2×C4—972no—
392C2×C4—972no—
393C2×C4—972no—
394C2×C4—972no—
395C2×C4—972no—
396C2×C4—3884no—
397C2×C4—3884no—
398C2×C4—1948no—
399C2×C4—1948no—
400C2×C4—1941no—
401C2×C4—1941no—
402C2×C4—1941no—
403C2×C4—1941no—
404C2×C4—1941no—
405C2×C4—1941no—
406C2×C4—1941no—
407C2×C4—1941no—
408C2×C4—1941no—
409C2×C4—1941no—
410C2×C4—1941no—
411C2×C4—1941no—
412C2×C4C2×C4×D9712yes—
413C2×C4D4×D9712yes—
414C2×C4#1309551012yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—3881no—
429C22—3881no—
430C22—3881no—
431C22—3881no—
432C22—1942no—
433C22—1942no—
434C22—1942no—
435C22#1569077014yescomplemented
436C22C22×C4×D9711yescharacteristic
437C22C2×D4×D9711yescharacteristic
438C22C2×D4×D9711yescharacteristic
439C22#41×D9711yescharacteristic
440C22#41×D9711yescharacteristic
441C22C2×#1309551011yescharacteristic
442C22C2×#1309551011yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—974no—
449C4—974no—
450C4—3881no—
451C4—3881no—
452C4—1942no—
453C4—1942no—
454C4—1942no—
455C2—24no—
456C2—42no—
457C2—3882no—
458C2C4×D4×D9711yescharacteristic
459C2C2×#1569077011yescharacteristic
460C2C4×#1309551011yescharacteristic
461C2#121×D9711yescharacteristic
462C2#2026124111yescharacteristic
463C2#2026126211yescharacteristic
464C2#2026128011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690782C2
hypercenter#386D97
hypocenterC97C4×#121
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic97C2semidirect
3. C2×#15690770C2semidirect
4. C2×#15690770C2semidirect
5. C22×C4×Dic97C2semidirect
6. #20260635C2semidirect
7. #20260632C2semidirect
8. #20260627C2semidirect
9. #20260623C2semidirect
10. #20260614C2semidirect
11. #20260593C2semidirect
12. #20260586C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. #20260582C2semidirect
15. #386×C97C2semidirect
22. #15690782C4semidirect
61. C2×C22⋊D97C2×C4semidirect
62. C2×C22⋊D97C2×C4semidirect
70. C22×Dic97C2×C4semidirect
71. C22×Dic97D4semidirect
86. #13095490C2×C4semidirect
89. C22⋊Dic97C2×C4semidirect
93. #13095488C2×C4semidirect
99. C4×Dic97D4semidirect
104. C2×D4×C97C2×C4semidirect
195. C22×C97C4×D4semidirect
220. Dic97C4×D4semidirect
221. Dic97#121semidirect
233. C97C4×#121semidirect
244. #386D97semidirect
300. C2×D4C4×D97semidirect
435. C22#15690770semidirect

Maximal subgroups

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

subgroupindex
C2×#156907822
C2×D4×Dic972
C2×#156907702
C22×C4×Dic972
#202606352
#202606322
#202606272
#202606232
#202606142
#202605932
#202605862
C22⋊C4×Dic972
#202605822
#386×C972
C4×#12197