Finite Group Databasebeta

← #27144978

Subgroups — #27144978

Order 12416.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×#15690769C211yesmaximal, characteristic, complemented
4#20260668C211yesmaximal, characteristic, complemented
5#20260663C211yesmaximal, characteristic, complemented
6#20260658C211yesmaximal, characteristic, complemented
7#20260657C211yesmaximal, characteristic, complemented
8#20260651C211yesmaximal, characteristic, complemented
9#20260650C211yesmaximal, characteristic, complemented
10#20260649C211yesmaximal, characteristic, complemented
11#20260643C211yesmaximal, characteristic
12C4×#13095490C211yesmaximal, characteristic, complemented
13C4×#13095488C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
15C2×C4×D4×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C4×C22⋊D97—24no—
20C4×C22⋊D97—24no—
21#15690769—24no—
22C2×C4⋊Dic97C2211yescharacteristic
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×C22⋊Dic97C2211yescharacteristic
33C2×C22⋊Dic97C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C22×D4×C97C2211yescharacteristic
48C23×C4×C97C2211yescharacteristic
49C23×C4×C97C2211yescharacteristic
50C4×D4×C97—24no—
51C2×C4⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C22⋊C4×C97C2211yescharacteristic
54C2×C42×C97C2211yescharacteristic
55C23×D97C2311yescharacteristic
56C2×C22⋊D97—22no—
57C2×C22⋊D97—22no—
58C2×C22⋊D97—22no—
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97—22no—
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C2×C4×D97—22no—
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C4⋊Dic97—22no—
73C4⋊Dic97—22no—
74#13095490—22no—
75#13095490—22no—
76#13095490—22no—
77#13095490—22no—
78C22⋊Dic97—22no—
79C22⋊Dic97—22no—
80C22⋊Dic97—22no—
81C22⋊Dic97—22no—
82C22⋊Dic97—22no—
83C22⋊Dic97—22no—
84#13095488—22no—
85#13095488—22no—
86#13095488—22no—
87#13095488—22no—
88C4×Dic97—22no—
89C4×Dic97—22no—
90C4×Dic97D414yescomplemented
91C24×C97C2311yescharacteristic
92C24×C97C2311yescharacteristic
93C2×D4×C97—22no—
94C2×D4×C97—22no—
95C2×D4×C97—22no—
96C22×C4×C97—22no—
97C22×C4×C97—22no—
98C22×C4×C97—22no—
99C22×C4×C97—22no—
100C22×C4×C97—22no—
101C22×C4×C97D414yescomplemented
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C4⋊C4×C97—22no—
108C22⋊C4×C97—22no—
109C22⋊C4×C97—22no—
110C42×C97—22no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C23×C97—21no—
122C23×C97—21no—
123C23×C97C2×D412yes—
124C23×C97#4112yes—
125C23×C97C2411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—21no—
131C22×D97—21no—
132C22×D97#4112yes—
133D4×C97—28no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—28no—
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×C97C2×D412yes—
147C2×C4×C97C2×D412yes—
148C2×C4×C97C2×D412yes—
149C2×C4×C97#4112yes—
150C2×C4×C97#4112yes—
151C22⋊D97—28no—
152C22⋊D97—28no—
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×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—21no—
171C2×Dic97C2×D412yes—
172C2×Dic97C2×D412yes—
173C2×Dic97#4112yes—
174C4×D97—44no—
175C22×C97—22no—
176C22×C97—22no—
177C22×C97—22no—
178C22×C97—22no—
179C22×C97—22no—
180C22×C97—22no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97C22×D411yescharacteristic
190C22×C97C22×D411yescharacteristic
191C22×C97C2×#4111yescharacteristic
192C22×C97C2×#4111yescharacteristic
193C22×C97C2×#4111yescharacteristic
194C22×C97C2×#4111yescharacteristic
195C22×C97C2×#4111yescharacteristic
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—41no—
206C4×C97—41no—
207C4×C97#12114yes—
208Dic97—22no—
209Dic97—22no—
210Dic97—22no—
211Dic97—41no—
212Dic97—41no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—42no—
216C2×C97—42no—
217C2×C97C2×#12111yescharacteristic
218C2×C97#52511yescharacteristic
219C2×C97#52511yescharacteristic
220C2×C97#52511yescharacteristic
221C2×C97#52911yescharacteristic
222C2×C97#52911yescharacteristic
223C2×C97#53211yescharacteristic
224D97—42no—
225#2392—971nomaximal, Sylow
226C97#239211yescharacteristic, complemented, Sylow
227C2×C4×D4—971no—
228C2×C4×D4—971no—
229C2×C4×D4D9711yescharacteristic, complemented
230C2×#117—971no—
231#398—971no—
232#397—971no—
233#396—971no—
234#396—971no—
235#394—971no—
236#394—971no—
237#394—971no—
238#392—971no—
239C4×C4⋊C4—971no—
240C4×C22⋊C4—971no—
241C4×C22⋊C4—971no—
242C22×D4—971no—
243C22×D4—971no—
244C22×D4C2×D9711yescharacteristic
245C23×C4—971no—
246C23×C4C2×D9711yescharacteristic
247C23×C4C2×D9711yescharacteristic
248C4×D4—24no—
249C4×D4—1944no—
250C4×D4—1944no—
251#117—1944no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—971no—
257C2×C4⋊C4—971no—
258C2×C4⋊C4C2×D9711yescharacteristic
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⋊C4—971no—
266C2×C22⋊C4—971no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4C2×D9711yescharacteristic
270C2×C22⋊C4C2×D9711yescharacteristic
271C2×C42—971no—
272C2×C42—971no—
273C2×C42—971no—
274C2×C42C2×D9711yescharacteristic
275#95—971no—
276#95—971no—
277#95—971no—
278#95—971no—
279#95—971no—
280#95—971no—
281C24—971no—
282C24C22×D9711yescharacteristic
283C24C22×D9711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—1942no—
288C2×D4—1942no—
289C2×D4—1942no—
290C2×D4—1942no—
291C2×D4—1942no—
292C2×D4—1942no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—971no—
299C22×C4—971no—
300C22×C4—971no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—971no—
304C22×C4—971no—
305C22×C4—1942no—
306C22×C4—1942no—
307C22×C4—1942no—
308C22×C4C22⋊D9714yescomplemented
309C22×C4C22×D9711yescharacteristic
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312C22×C4C22×D9711yescharacteristic
313C22×C4C22×D9711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—1942no—
316C4⋊C4—1942no—
317C4⋊C4—1942no—
318C4⋊C4—1942no—
319C4⋊C4—1942no—
320C4⋊C4—1942no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—1942no—
324C22⋊C4—1942no—
325C22⋊C4—1942no—
326C22⋊C4—1942no—
327C22⋊C4—1942no—
328C22⋊C4—1942no—
329C22⋊C4—1942no—
330C22⋊C4—1942no—
331C22⋊C4—1942no—
332C22⋊C4—1942no—
333C42—22no—
334C42—974no—
335C42—1942no—
336C42—1942no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—972no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23—1941no—
355C23—1941no—
356C23C2×C22⋊D9712yes—
357C23#1309551112yes—
358C23C23×D9711yescharacteristic
359D4—28no—
360D4—1948no—
361D4—1948no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
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—3884no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
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×C4—1941no—
395C2×C4—1941no—
396C2×C4D4×D9712yes—
397C2×C4C2×C22⋊D9712yes—
398C2×C4C2×C22⋊D9712yes—
399C2×C4#1309551012yes—
400C2×C4#1309551112yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—3881no—
416C22—3881no—
417C22—3881no—
418C22—3881no—
419C22—1942no—
420C22—1942no—
421C22—1942no—
422C22C2×D4×D9711yescharacteristic
423C22C22×C22⋊D9711yescharacteristic
424C22#41×D9711yescharacteristic
425C22#41×D9711yescharacteristic
426C22C2×#1309551011yescharacteristic
427C22C2×#1309551111yescharacteristic
428C22C2×#1309551111yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—3881no—
434C4—3881no—
435C4—3881no—
436C4—3881no—
437C4—1942no—
438C4—1942no—
439C4—1942no—
440C4#1569078214yes—
441C2—42no—
442C2—42no—
443C2—3882no—
444C2C2×#1569078211yescharacteristic
445C2#2026124211yescharacteristic
446C2#2026124411yescharacteristic
447C2#2026124611yescharacteristic
448C2#2026126511yescharacteristic
449C2#2026126711yescharacteristic
450C2#2026128211yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC2×C4×D4×C97C2
prime radical#20260643C2
hypercenterC2×C4×D4D97
hypocenterC97#2392
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D97C2semidirect
2. C2×C4×C22⋊D97C2semidirect
3. C2×#15690769C2semidirect
4. #20260668C2semidirect
5. #20260663C2semidirect
6. #20260658C2semidirect
7. #20260657C2semidirect
8. #20260651C2semidirect
9. #20260650C2semidirect
10. #20260649C2semidirect
12. C4×#13095490C2semidirect
13. C4×#13095488C2semidirect
14. C4×C22⋊Dic97C2semidirect
15. C2×C4×D4×C97C2semidirect
90. C4×Dic97D4semidirect
101. C22×C4×C97D4semidirect
226. C97#2392semidirect
229. C2×C4×D4D97semidirect
308. C22×C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D972
C2×#156907692
#202606682
#202606632
#202606582
#202606572
#202606512
#202606502
#202606492
#202606432
C4×#130954902
C4×#130954882
C4×C22⋊Dic972
C2×C4×D4×C972
#239297