Finite Group Databasebeta

← #27144980

Subgroups — #27144980

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×D388C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
4C2×#15690767C211yesmaximal, characteristic, complemented
5#20260669C211yesmaximal, characteristic, complemented
6#20260664C211yesmaximal, characteristic, complemented
7#20260659C211yesmaximal, characteristic, complemented
8#20260658C211yesmaximal, characteristic, complemented
9#20260653C211yesmaximal, characteristic, complemented
10#20260650C211yesmaximal, characteristic, complemented
11#20260650C211yesmaximal, characteristic, complemented
12#20260644C211yesmaximal, characteristic
13C4×#13095490C211yesmaximal, characteristic, complemented
14C4×#13095488C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C4×D388—24no—
22C4×C22⋊D97—24no—
23C4×C22⋊D97—24no—
24#15690767—24no—
25C2×C4⋊Dic97C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×#13095490C2211yescharacteristic
30C2×#13095490C2211yescharacteristic
31C2×#13095490C2211yescharacteristic
32C2×C22⋊Dic97C2211yescharacteristic
33C2×C22⋊Dic97C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×#13095488C2211yescharacteristic
40C2×#13095488C2211yescharacteristic
41C2×#13095488C2211yescharacteristic
42C2×C4×Dic97C2211yescharacteristic
43C2×C4×Dic97C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47#15690717C2211yescharacteristic
48C23×C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52C2×C42×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54#95×C97C2211yescharacteristic
55C23×D97C2311yescharacteristic
56C23×D97C2311yescharacteristic
57C2×D388—22no—
58C2×D388—22no—
59C2×D388—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97—22no—
62C2×C22⋊D97—22no—
63C2×C22⋊D97—22no—
64C2×C22⋊D97—22no—
65C2×C22⋊D97—22no—
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97D414yescomplemented
78C4⋊Dic97—22no—
79#13095490—22no—
80#13095490—22no—
81#13095490—22no—
82#13095490—22no—
83#13095490—22no—
84#13095490—22no—
85C22⋊Dic97—22no—
86C22⋊Dic97—22no—
87#13095488—22no—
88#13095488—22no—
89#13095488—22no—
90#13095488—22no—
91#13095488—22no—
92#13095488—22no—
93#13095488—22no—
94#13095488—22no—
95C4×Dic97—22no—
96C4×Dic97D414yescomplemented
97C24×C97C2311yescharacteristic
98C22×C4×C97—22no—
99C22×C4×C97—22no—
100C22×C4×C97—22no—
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C22⋊C4×C97—22no—
108C22⋊C4×C97—22no—
109C42×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#4112yes—
118C23×C97C2411yescharacteristic
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×D97—21no—
131C22×D97C2×D412yes—
132C22×D97#4112yes—
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—21no—
146C2×C4×C97—21no—
147C2×C4×C97—44no—
148C2×C4×C97C2×D412yes—
149C2×C4×C97C2×D412yes—
150C2×C4×C97#4112yes—
151C2×C4×C97#4112yes—
152C22⋊D97—28no—
153C22⋊D97—28no—
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×Dic97C2×D412yes—
169C2×Dic97C2×D412yes—
170C2×Dic97C2×D412yes—
171C2×Dic97#4112yes—
172D388—28no—
173C4×D97—28no—
174C4×D97—44no—
175C22×C97—22no—
176C22×C97—22no—
177C22×C97—22no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97C22×D411yescharacteristic
183C22×C97C22×D411yescharacteristic
184C22×C97C2×#4111yescharacteristic
185C22×C97C2×#4111yescharacteristic
186C22×C97C2×#4111yescharacteristic
187C22×C97C2×#4111yescharacteristic
188C22×C97C2×#4111yescharacteristic
189C2×D97—22no—
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C4×C97—22no—
204C4×C97—41no—
205C4×C97—41no—
206C4×C97—41no—
207C4×C97—41no—
208C4×C97#12114yes—
209Dic97—22no—
210Dic97—22no—
211Dic97—22no—
212Dic97—22no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—42no—
216C2×C97C2×#12111yescharacteristic
217C2×C97#52511yescharacteristic
218C2×C97#52511yescharacteristic
219C2×C97#52511yescharacteristic
220C2×C97#52911yescharacteristic
221C2×C97#52911yescharacteristic
222C2×C97#53211yescharacteristic
223D97—42no—
224D97—42no—
225#2392—971nomaximal, Sylow
226C97#239211yescharacteristic, complemented, Sylow
227C2×C4×D4—971no—
228C2×C4×D4—971no—
229C2×C4×D4—971no—
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⋊C4D9711yescharacteristic, complemented
242C22×D4—971no—
243C22×D4—971no—
244C22×D4—971no—
245C23×C4—971no—
246C23×C4—971no—
247C23×C4C2×D9711yescharacteristic
248C4×D4—1944no—
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⋊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⋊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×C42C2×D9711yescharacteristic
274C2×C42C2×D9711yescharacteristic
275#95—971no—
276#95—971no—
277#95—971no—
278#95—971no—
279#95C2×D9711yescharacteristic
280#95C2×D9711yescharacteristic
281C24—971no—
282C24—971no—
283C24C22×D9711yescharacteristic
284C2×D4—1942no—
285C2×D4—1942no—
286C2×D4—1942no—
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—971no—
297C22×C4—971no—
298C22×C4—971no—
299C22×C4—971no—
300C22×C4—974no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—1942no—
304C22×C4—1942no—
305C22×C4—1942no—
306C22×C4—1942no—
307C22×C4—1942no—
308C22×C4C22×D9711yescharacteristic
309C22×C4C22×D9711yescharacteristic
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312C22×C4C22×D9711yescharacteristic
313C22×C4C22×D9711yescharacteristic
314C4⋊C4—1942no—
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—22no—
335C42—974no—
336C42—1942no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—972no—
344C23—972no—
345C23—1941no—
346C23—1941no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23—1941no—
355C23—1941no—
356C23—1941no—
357C23#1309551112yes—
358C23C23×D9711yescharacteristic
359D4—1948no—
360D4—1948no—
361D4—1948no—
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—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—972no—
378C2×C4—972no—
379C2×C4—972no—
380C2×C4—972no—
381C2×C4—3884no—
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×C4—1948no—
397C2×C4D4×D9712yes—
398C2×C4D4×D9712yes—
399C2×C4#1309551112yes—
400C2×C4#1309551112yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—3881no—
415C22—3881no—
416C22—1942no—
417C22—1942no—
418C22—1942no—
419C22—1942no—
420C22—1942no—
421C22—1942no—
422C22C2×D4×D9711yescharacteristic
423C22C2×D4×D9711yescharacteristic
424C22#41×D9711yescharacteristic
425C22#41×D9711yescharacteristic
426C22C2×#1309551111yescharacteristic
427C22C2×#1309551111yescharacteristic
428C22C2×#1309551111yescharacteristic
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—3881no—
435C4—3881no—
436C4—1942no—
437C4—1942no—
438C4—1942no—
439C4—1942no—
440C4#1569078314yes—
441C2—42no—
442C2—3882no—
443C2—3882no—
444C2C2×#1569078311yescharacteristic
445C2#2026124411yescharacteristic
446C2#2026124411yescharacteristic
447C2#2026124411yescharacteristic
448C2#2026126311yescharacteristic
449C2#2026126911yescharacteristic
450C2#2026128211yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC4×C22⋊C4×C97C2
prime radical#20260644C2
hypercenterC4×C22⋊C4D97
hypocenterC97#2392
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D388C2semidirect
2. C2×C4×C22⋊D97C2semidirect
3. C2×C4×C22⋊D97C2semidirect
4. C2×#15690767C2semidirect
5. #20260669C2semidirect
6. #20260664C2semidirect
7. #20260659C2semidirect
8. #20260658C2semidirect
9. #20260653C2semidirect
10. #20260650C2semidirect
11. #20260650C2semidirect
13. C4×#13095490C2semidirect
14. C4×#13095488C2semidirect
15. C4×C22⋊C4×C97C2semidirect
77. C2×C4×D97D4semidirect
96. C4×Dic97D4semidirect
226. C97#2392semidirect
241. C4×C22⋊C4D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3882
C2×C4×C22⋊D972
C2×#156907672
#202606692
#202606642
#202606592
#202606582
#202606532
#202606502
#202606442
C4×#130954902
C4×#130954882
C4×C22⋊C4×C972
#239297