Finite Group Databasebeta

← #27158252

Subgroups — #27158252

Order 12416.

660 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690889C211yesmaximal, characteristic, complemented
2C22×C4⋊#87604C211yesmaximal, characteristic, complemented
3#20261685C212yesmaximal, complemented
4#20261684C212yesmaximal, complemented
5#20261683C212yesmaximal
6#20261679C211yesmaximal, characteristic, complemented
7#20261677C211yesmaximal, characteristic, complemented
8#20261677C211yesmaximal, characteristic, complemented
9C4×C4⋊#87604C211yesmaximal, characteristic, complemented
10C22⋊C4×#87604C212yesmaximal, complemented
11C4×D4×D97C211yesmaximal, characteristic
12C23×#87604C2212yes—
13#15690889—24no—
14C2×C4⋊#87604—24no—
15C2×C4⋊#87604C2211yescharacteristic
16C2×C4⋊#87604C2211yescharacteristic
17C2×C4⋊#87604C2211yescharacteristic
18C2×C4⋊#87604C2212yes—
19C2×C4⋊#87604C2211yescharacteristic
20C2×C22⋊#87604C2212yes—
21C2×C22⋊#87604C2212yes—
22C2×C22⋊#87604C2212yes—
23C2×C4×#87604C2212yes—
24C2×C4×#87604C2212yes—
25(C2×#87604)⋊C4C2212yes—
26(C2×#87604)⋊C4C2212yes—
27(C2×#87604)⋊C4C2212yes—
28(C2×#87604)⋊C4C2212yes—
29(C2×#87604)⋊C4C2212yes—
30C2×D4×D97C2211yescharacteristic
31C22×C4×D97C2211yescharacteristic
32C22×C4×D97C2211yescharacteristic
33C4×D388C411yescharacteristic, complemented
34D4×Dic97C411yescharacteristic, complemented
35#15690772C411yescharacteristic, complemented
36C4×C22⋊D97C411yescharacteristic, complemented
37C4×C22⋊D97C411yescharacteristic, complemented
38#15690770C411yescharacteristic, complemented
39#15690770C411yescharacteristic, complemented
40C4⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97C2211yescharacteristic
42C22⋊C4×D97C2211yescharacteristic
43C42×D97C2211yescharacteristic
44C4×D4×C97C411yescharacteristic, complemented
45C22×#87604—24no—
46C22×#87604—24no—
47C22×#87604—42no—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C22×#87604C2312yes—
51C22×#87604C2312yes—
52C4⋊#87604—24no—
53C4⋊#87604—22no—
54C4⋊#87604—22no—
55C4⋊#87604—22no—
56C4⋊#87604D414yescomplemented
57C22⋊#87604—24no—
58C22⋊#87604—24no—
59C22⋊#87604—42no—
60C4×#87604—24no—
61C4×#87604—42no—
62C23×D97C2311yescharacteristic
63C23×D97C2311yescharacteristic
64C2×D388C2×C411yescharacteristic
65C2×C22⋊D97C2×C411yescharacteristic
66C2×C22⋊D97C2×C411yescharacteristic
67D4×D97—22no—
68D4×D97—22no—
69C22×Dic97C2×C411yescharacteristic
70C22×Dic97C2×C411yescharacteristic, complemented
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97C2×C411yescharacteristic
76C2×C4×D97C2×C411yescharacteristic
77C2×C4×D97C2×C411yescharacteristic, complemented
78C2×C4×D97C2×C411yescharacteristic, complemented
79C2×C4×D97C2311yescharacteristic
80C2×C4×D97C2311yescharacteristic
81C2×C4×D97C2311yescharacteristic
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C4⋊Dic97C2×C411yescharacteristic, complemented
85#13095490C2×C411yescharacteristic, complemented
86#13095490C2×C411yescharacteristic, complemented
87C22⋊Dic97C2×C411yescharacteristic
88C22⋊Dic97C2×C411yescharacteristic, complemented
89#13095488C2×C411yescharacteristic
90#13095488C2×C411yescharacteristic
91#13095488C2×C411yescharacteristic, complemented
92#13095488C2×C411yescharacteristic, complemented
93C4×Dic97C2×C411yescharacteristic, complemented
94C4×Dic97C2×C411yescharacteristic, complemented
95C4×Dic97C2×C411yescharacteristic, complemented
96C2×D4×C97C2×C411yescharacteristic
97C22×C4×C97C2×C411yescharacteristic
98C22×C4×C97C2×C411yescharacteristic, complemented
99C4⋊C4×C97C2×C411yescharacteristic, complemented
100C22⋊C4×C97C2×C411yescharacteristic
101C22⋊C4×C97C2×C411yescharacteristic, complemented
102C42×C97C2×C411yescharacteristic, complemented
103C23×C97C22×C411yescharacteristic
104C23×C97C22×C411yescharacteristic
105C22×D97—21no—
106C22×D97—21no—
107C22×D97—21no—
108C22×D97—21no—
109C22×D97—21no—
110C22×D97—21no—
111C22×D97—21no—
112C22×D97—21no—
113C22×D97—21no—
114C22×D97—21no—
115C22×D97C22×C411yescharacteristic
116C22×D97C22×C411yescharacteristic
117C22×D97#4112yes—
118C22×D97#4112yes—
119C22×D97C2411yescharacteristic
120C2×#87604—22no—
121C2×#87604—22no—
122C2×#87604—22no—
123C2×#87604—22no—
124C2×#87604—22no—
125C2×#87604—22no—
126C2×#87604—22no—
127C2×#87604—22no—
128C2×#87604—22no—
129C2×#87604—22no—
130C2×#87604—22no—
131C2×#87604—48no—
132C2×#87604C2×D414yes—
133D4×C97—22no—
134C2×C4×C97—22no—
135C2×C4×C97—22no—
136C2×C4×C97C22×C411yescharacteristic
137C2×C4×C97C22×C411yescharacteristic
138C2×C4×C97C22×C411yescharacteristic
139C2×C4×C97C22×C411yescharacteristic
140C2×C4×C97C22×C411yescharacteristic
141C22⋊D97—22no—
142C22⋊D97—22no—
143C2×Dic97—22no—
144C2×Dic97—22no—
145C2×Dic97C22×C411yescharacteristic
146C2×Dic97C22×C411yescharacteristic
147C2×Dic97C22×C411yescharacteristic
148C2×Dic97C22×C411yescharacteristic
149C2×Dic97C22×C411yescharacteristic
150D388—22no—
151C4×D97—21no—
152C4×D97—21no—
153C4×D97—21no—
154C4×D97—21no—
155C4×D97—21no—
156C4×D97—21no—
157C4×D97—21no—
158C4×D97—21no—
159C4×D97—22no—
160C4×D97—22no—
161C4×D97—22no—
162C4×D97—22no—
163C4×D97C2×D412yes—
164C4×D97#4112yes—
165C22×C97—21no—
166C22×C97—21no—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97#11712yescomplemented
171C22×C97C23×C411yescharacteristic
172C2×D97—21no—
173C2×D97—21no—
174C2×D97—22no—
175C2×D97—22no—
176C2×D97—22no—
177C2×D97—22no—
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97#11712yescomplemented
186C2×D97C22×D411yescharacteristic
187C2×D97C2×#4111yescharacteristic
188C2×D97C2×#4111yescharacteristic
189C2×D97C2×#4111yescharacteristic
190C2×D97H(5,𝔽2)11yescharacteristic
191C2×D972−1+411yescharacteristic
192#87604—24no—
193#87604—42no—
194#87604—42no—
195#87604—42no—
196C4×C97—21no—
197C4×C97—21no—
198C4×C97—21no—
199C4×C97—21no—
200C4×C97C4×D412yes—
201Dic97—21no—
202Dic97—21no—
203Dic97—21no—
204Dic97—21no—
205Dic97C4×D412yes—
206C2×C97—22no—
207C2×C97—41no—
208C2×C97C2×#11711yescharacteristic
209C2×C97C2×C4×D411yescharacteristic
210C2×C97#52011yescharacteristic
211D97—22no—
212D97—41no—
213D97#54611yescharacteristic
214D97#54811yescharacteristic
215D97#55211yescharacteristic
216D97#55311yescharacteristic
217#2352—971nomaximal, Sylow
218C97#235211yescharacteristic, complemented, Sylow
219C2×C4×D4—971no—
220C2×#117—971no—
221C22×C4⋊C4—971no—
222#388—972no—
223#387—972no—
224#386—972no—
225#382—971no—
226#380—971no—
227#380—971no—
228C4×C4⋊C4—971no—
229C4×C22⋊C4—972no—
230C22×D4—971no—
231C23×C4—971no—
232C23×C4—971no—
233C23×C4—972no—
234C4×D4—971no—
235C4×D4—971no—
236C4×D4—971no—
237C4×D4—971no—
238C4×D4—971no—
239C4×D4—971no—
240C4×D4—971no—
241C4×D4#8760411yescharacteristic, complemented
242#117—1944no—
243C2×C4⋊C4—971no—
244C2×C4⋊C4—971no—
245C2×C4⋊C4—971no—
246C2×C4⋊C4—972no—
247C2×C4⋊C4—971no—
248C2×C4⋊C4—971no—
249C2×C4⋊C4—1944no—
250C2×C22⋊C4—971no—
251C2×C22⋊C4—971no—
252C2×C22⋊C4—972no—
253C2×C22⋊C4—972no—
254C2×C22⋊C4—972no—
255C2×C42—971no—
256C2×C42—972no—
257C2×C42—972no—
258#95—972no—
259#95—972no—
260#95—972no—
261#95—972no—
262#95—972no—
263C24—971no—
264C24—971no—
265C2×D4—971no—
266C2×D4—971no—
267C2×D4—971no—
268C2×D4—1942no—
269C2×D4—1942no—
270C2×D4C2×#8760411yescharacteristic
271C22×C4—971no—
272C22×C4—972no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—971no—
283C22×C4—972no—
284C22×C4—972no—
285C22×C4—972no—
286C22×C4—3882no—
287C22×C4—1942no—
288C22×C4—1942no—
289C22×C4—1942no—
290C22×C4—1942no—
291C22×C4—1944no—
292C22×C4—1944no—
293C22×C4C2×#8760411yescharacteristic
294C22×C4C2×#8760411yescharacteristic, complemented
295C4⋊C4—974no—
296C4⋊C4—971no—
297C4⋊C4—971no—
298C4⋊C4—971no—
299C4⋊C4—1942no—
300C4⋊C4—1942no—
301C4⋊C4—1944no—
302C4⋊C4—1942no—
303C4⋊C4C2×#8760411yescharacteristic, complemented
304C22⋊C4—971no—
305C22⋊C4—971no—
306C22⋊C4—971no—
307C22⋊C4—971no—
308C22⋊C4—971no—
309C22⋊C4—971no—
310C22⋊C4—3882no—
311C22⋊C4—1944no—
312C22⋊C4—1944no—
313C22⋊C4C2×#8760411yescharacteristic
314C22⋊C4C2×#8760411yescharacteristic, complemented
315C42—971no—
316C42—971no—
317C42—971no—
318C42—3882no—
319C42—1944no—
320C42C2×#8760411yescharacteristic, complemented
321C23—971no—
322C23—972no—
323C23—972no—
324C23—971no—
325C23—971no—
326C23—1941no—
327C23—1941no—
328C23—1941no—
329C23—1941no—
330C23—1941no—
331C23—1941no—
332C23—1941no—
333C23—1941no—
334C23—1941no—
335C23—1941no—
336C23C22×#8760411yescharacteristic
337C23C22×#8760411yescharacteristic
338D4—22no—
339D4—1942no—
340D4—1942no—
341D4—1942no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—971no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—974no—
348C2×C4—971no—
349C2×C4—971no—
350C2×C4—971no—
351C2×C4—971no—
352C2×C4—3888no—
353C2×C4—1942no—
354C2×C4—1942no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1942no—
364C2×C4—1942no—
365C2×C4—1942no—
366C2×C4—1942no—
367C2×C4—1942no—
368C2×C4—1942no—
369C2×C4—1942no—
370C2×C4—1942no—
371C2×C4—1942no—
372C2×C4—1942no—
373C2×C4—1942no—
374C2×C4—1942no—
375C2×C4—1942no—
376C2×C4—1942no—
377C2×C4—1942no—
378C2×C4C22×#8760411yescharacteristic
379C2×C4C22×#8760411yescharacteristic
380C2×C4C22×#8760411yescharacteristic
381C2×C4C22×#8760411yescharacteristic
382C2×C4C22×#8760411yescharacteristic
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—971no—
389C22—971no—
390C22—971no—
391C22—971no—
392C22—971no—
393C22—971no—
394C22—972no—
395C22—3881no—
396C22—3881no—
397C22—3881no—
398C22—3881no—
399C22—1942no—
400C22—1942no—
401C22—1942no—
402C22—1942no—
403C22—1941no—
404C22—1941no—
405C22—1941no—
406C22—1941no—
407C22—1941no—
408C22#1569088912yescomplemented
409C22C23×#8760411yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—972no—
415C4—3882no—
416C4—3882no—
417C4—3882no—
418C4—1944no—
419C4—1941no—
420C4—1941no—
421C4—1941no—
422C4—1941no—
423C4D4×#8760412yes—
424C2—22no—
425C2—41no—
426C2—971no—
427C2—971no—
428C2—971no—
429C2—971no—
430C2—3881no—
431C2—1942no—
432C2C2×#1569088911yescharacteristic
433C2C2×D4×#8760411yescharacteristic
434C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC4×D4×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenterC4×D4#87604
hypocenterC97#2352
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690889C2semidirect
2. C22×C4⋊#87604C2semidirect
3. #20261685C2semidirect
4. #20261684C2semidirect
6. #20261679C2semidirect
7. #20261677C2semidirect
8. #20261677C2semidirect
9. C4×C4⋊#87604C2semidirect
10. C22⋊C4×#87604C2semidirect
33. C4×D388C4semidirect
34. D4×Dic97C4semidirect
35. #15690772C4semidirect
36. C4×C22⋊D97C4semidirect
37. C4×C22⋊D97C4semidirect
38. #15690770C4semidirect
39. #15690770C4semidirect
44. C4×D4×C97C4semidirect
56. C4⋊#87604D4semidirect
70. C22×Dic97C2×C4semidirect
77. C2×C4×D97C2×C4semidirect
78. C2×C4×D97C2×C4semidirect
84. C4⋊Dic97C2×C4semidirect
85. #13095490C2×C4semidirect
86. #13095490C2×C4semidirect
88. C22⋊Dic97C2×C4semidirect
91. #13095488C2×C4semidirect
92. #13095488C2×C4semidirect
93. C4×Dic97C2×C4semidirect
94. C4×Dic97C2×C4semidirect
95. C4×Dic97C2×C4semidirect
98. C22×C4×C97C2×C4semidirect
99. C4⋊C4×C97C2×C4semidirect
101. C22⋊C4×C97C2×C4semidirect
102. C42×C97C2×C4semidirect
170. C22×C97#117semidirect
185. C2×D97#117semidirect
218. C97#2352semidirect
241. C4×D4#87604semidirect
294. C22×C4C2×#87604semidirect
303. C4⋊C4C2×#87604semidirect
314. C22⋊C4C2×#87604semidirect
320. C42C2×#87604semidirect
408. C22#15690889semidirect

Maximal subgroups

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

subgroupindex
C2×#156908892
C22×C4⋊#876042
#202616852
#202616842
#202616832
#202616792
#202616772
C4×C4⋊#876042
C22⋊C4×#876042
C4×D4×D972
#235297