Finite Group Databasebeta

← #27144775

Subgroups — #27144775

Order 12416.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690783C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×#15690767C211yesmaximal, characteristic, complemented
4#20260637C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260632C211yesmaximal, characteristic, complemented
8#20260628C211yesmaximal, characteristic, complemented
9#20260626C211yesmaximal, characteristic, complemented
10#20260614C211yesmaximal, characteristic, complemented
11#20260613C211yesmaximal, characteristic, complemented
12C4×#13095490C211yesmaximal, characteristic, complemented
13#20260582C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690783C418yescomplemented
22C4×C22⋊D97—24no—
23#15690767—24no—
24C2×C4⋊Dic97C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41C2×C4×Dic97C2211yescharacteristic
42C2×C4×Dic97C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C23×C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C42×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C23×D97C2311yescharacteristic
56C2×D388—41no—
57C2×D388—41no—
58C2×D388C2×C414yescomplemented
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97—41no—
62C2×C22⋊D97—41no—
63C2×C22⋊D97C2×C414yes—
64C2×C22⋊D97C2×C414yescomplemented
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—41no—
76C2×C4×D97C2×C414yescomplemented
77C4⋊Dic97—41no—
78#13095490—22no—
79#13095490—41no—
80#13095490C2×C414yescomplemented
81#13095490D414yescomplemented
82C22⋊Dic97—22no—
83C22⋊Dic97—41no—
84#13095488—22no—
85#13095488—22no—
86#13095488—22no—
87#13095488—22no—
88#13095488—22no—
89#13095488—22no—
90#13095488—41no—
91#13095488C2×C414yescomplemented
92C4×Dic97—22no—
93C4×Dic97—22no—
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×C414yescomplemented
108C42×C97—22no—
109C42×C97—41no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97C22×C412yes—
117C23×C97C2411yescharacteristic
118C22×D97—21no—
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97C22×C412yes—
131C22×D97C22×C412yes—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
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—44no—
146C2×C4×C97C22×C412yes—
147C2×C4×C97C22×C412yes—
148C2×C4×C97C2×D412yes—
149C2×C4×C97#4112yes—
150C2×C4×C97#4112yes—
151C22⋊D97—28no—
152C22⋊D97—44no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97—21no—
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×Dic97C22×C412yes—
167C2×Dic97C22×C412yes—
168C2×Dic97C2×D412yes—
169C2×Dic97C2×D412yes—
170C2×Dic97#4112yes—
171D388—44no—
172C4×D97—28no—
173C4×D97—44no—
174C22×C97—22no—
175C22×C97—22no—
176C22×C97—22no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97C23×C411yescharacteristic
182C22×C97C22×D411yescharacteristic
183C22×C97C2×#4111yescharacteristic
184C22×C97C2×#4111yescharacteristic
185C22×C97C2×#4111yescharacteristic
186C22×C97H(5,𝔽2)11yescharacteristic
187C22×C97H(5,𝔽2)11yescharacteristic
188C2×D97—22no—
189C2×D97—22no—
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C4×C97—22no—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—22no—
206C4×C97—22no—
207C4×C97—41no—
208Dic97—22no—
209Dic97—22no—
210Dic97—41no—
211Dic97—41no—
212Dic97—41no—
213Dic97C4×D414yes—
214C2×C97—42no—
215C2×C97C2×C4×D411yescharacteristic
216C2×C97C4×#4111yescharacteristic
217C2×C97#51811yescharacteristic
218C2×C97#54611yescharacteristic
219C2×C97#55011yescharacteristic
220C2×C97#55311yescharacteristic
221C2×C97#55511yescharacteristic
222D97—42no—
223D97—42no—
224#2363—971nomaximal, Sylow
225C97#236311yescharacteristic, complemented, Sylow
226C2×#121—971no—
227C2×C4×D4—971no—
228C2×#117—971no—
229#390—971no—
230#390—971no—
231#390—971no—
232#389—971no—
233#388—971no—
234#388—971no—
235#386—971no—
236#386—971no—
237C4×C4⋊C4—971no—
238C4×C22⋊C4—971no—
239C4×C22⋊C4—971no—
240C4×C22⋊C4D9711yescharacteristic, complemented
241C22×D4—971no—
242C22×D4—971no—
243C22×D4—971no—
244C23×C4—971no—
245C23×C4—971no—
246C23×C4C2×D9711yescharacteristic
247#121—978no—
248C4×D4—1944no—
249#117—1944no—
250C2×C4⋊C4—971no—
251C2×C4⋊C4—971no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C22⋊C4—971no—
256C2×C22⋊C4—971no—
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—971no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4C2×D9711yescharacteristic
266C2×C22⋊C4C2×D9711yescharacteristic
267C2×C42—971no—
268C2×C42—971no—
269C2×C42—971no—
270C2×C42—971no—
271C2×C42C2×D9711yescharacteristic
272C2×C42C2×D9711yescharacteristic
273#95—971no—
274#95—971no—
275#95—971no—
276#95—971no—
277#95C2×D9711yescharacteristic
278#95C2×D9711yescharacteristic
279C24—971no—
280C24—971no—
281C24C22×D9711yescharacteristic
282C2×D4—974no—
283C2×D4—974no—
284C2×D4—974no—
285C2×D4—3881no—
286C2×D4—3881no—
287C2×D4—3881no—
288C2×D4—3881no—
289C2×D4—1942no—
290C2×D4—1942no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—974no—
298C22×C4—971no—
299C22×C4—971no—
300C22×C4—971no—
301C22×C4—3881no—
302C22×C4—1942no—
303C22×C4—1942no—
304C22×C4—1942no—
305C22×C4—1942no—
306C22×C4C22×D9711yescharacteristic
307C22×C4C22×D9711yescharacteristic
308C22×C4C22×D9711yescharacteristic
309C22×C4C22×D9711yescharacteristic
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312C4⋊C4—974no—
313C4⋊C4—974no—
314C4⋊C4—3881no—
315C4⋊C4—3881no—
316C4⋊C4—1942no—
317C22⋊C4—22no—
318C22⋊C4—974no—
319C22⋊C4—3881no—
320C22⋊C4—3881no—
321C22⋊C4—1942no—
322C22⋊C4—1942no—
323C22⋊C4—1942no—
324C22⋊C4—1942no—
325C22⋊C4—1942no—
326C22⋊C4—1942no—
327C22⋊C4—1942no—
328C22⋊C4C4×D9714yescomplemented
329C42—22no—
330C42—41no—
331C42—3881no—
332C42—1942no—
333C42—1942no—
334C42—1942no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—972no—
342C23—972no—
343C23—1941no—
344C23—1941no—
345C23—1941no—
346C23—1941no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23—1941no—
355C23C2×C4×D9712yes—
356C23C23×D9711yescharacteristic
357D4—3884no—
358D4—3884no—
359D4—1948no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—972no—
375C2×C4—972no—
376C2×C4—972no—
377C2×C4—972no—
378C2×C4—972no—
379C2×C4—3884no—
380C2×C4—1948no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4—1941no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4—1941no—
392C2×C4—1941no—
393C2×C4—1941no—
394C2×C4C2×C4×D9712yes—
395C2×C4C2×C4×D9712yes—
396C2×C4D4×D9712yes—
397C2×C4#1309551112yes—
398C2×C4#1309551112yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3881no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—1942no—
415C22—1942no—
416C22—1942no—
417C22—1942no—
418C22—1942no—
419C22—1942no—
420C22C22×C4×D9711yescharacteristic
421C22C2×D4×D9711yescharacteristic
422C22#41×D9711yescharacteristic
423C22C2×#1309551111yescharacteristic
424C22C2×#1309551111yescharacteristic
425C22#1569085511yescharacteristic
426C22#1569085611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—974no—
434C4—3881no—
435C4—3881no—
436C4—3881no—
437C4—1942no—
438C4—1942no—
439C2—42no—
440C2—3882no—
441C2—3882no—
442C2C4×D4×D9711yescharacteristic
443C2C4×#1309551111yescharacteristic
444C2#2026120711yescharacteristic
445C2#2026137511yescharacteristic
446C2#2026140611yescharacteristic
447C2#2026143411yescharacteristic
448C2#2026144811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC4×C22⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690783C2
hypercenterC4×C22⋊C4D97
hypocenterC97#2363
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D97C2semidirect
3. C2×#15690767C2semidirect
4. #20260637C2semidirect
5. #20260635C2semidirect
6. #20260635C2semidirect
7. #20260632C2semidirect
8. #20260628C2semidirect
9. #20260626C2semidirect
10. #20260614C2semidirect
11. #20260613C2semidirect
12. C4×#13095490C2semidirect
13. #20260582C2semidirect
14. #20260582C2semidirect
15. C4×C22⋊C4×C97C2semidirect
21. #15690783C4semidirect
58. C2×D388C2×C4semidirect
64. C2×C22⋊D97C2×C4semidirect
76. C2×C4×D97C2×C4semidirect
80. #13095490C2×C4semidirect
81. #13095490D4semidirect
91. #13095488C2×C4semidirect
107. C22⋊C4×C97C2×C4semidirect
225. C97#2363semidirect
240. C4×C22⋊C4D97semidirect
328. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907832
C2×C4×C22⋊D972
C2×#156907672
#202606372
#202606352
#202606322
#202606282
#202606262
#202606142
#202606132
C4×#130954902
#202605822
C4×C22⋊C4×C972
#236397