Finite Group Databasebeta

← #28393255

Subgroups — #28393255

Order 14464.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813600C211yesmaximal, characteristic
2C2×C4×Dic226C211yesmaximal, characteristic, complemented
3C2×#16813580C211yesmaximal, characteristic
4#20298502C211yesmaximal, characteristic, complemented
5#20298497C211yesmaximal, characteristic, complemented
6#20298495C211yesmaximal, characteristic, complemented
7#20298495C211yesmaximal, characteristic, complemented
8#20298490C211yesmaximal, characteristic
9#20298490C211yesmaximal, characteristic
10#20298489C211yesmaximal, characteristic
11#20298487C211yesmaximal, characteristic
12#20298479C211yesmaximal, characteristic, complemented
13#20298457C211yesmaximal, characteristic
14#20298449C211yesmaximal, characteristic, complemented
15#387×C113C211yesmaximal, characteristic, complemented
16C22×Dic226C2211yescharacteristic
17C22×C4×D113C2211yescharacteristic
18#16813600C418yescomplemented
19C4×Dic226—24no—
20#16813580—24no—
21C2×C4⋊Dic113C2211yescharacteristic
22C2×C4⋊Dic113C2211yescharacteristic
23C2×C4⋊Dic113C2211yescharacteristic
24C2×#14268626C2211yescharacteristic
25C2×#14268626C2211yescharacteristic
26C2×#14268626C2211yescharacteristic
27C2×#14268626C2211yescharacteristic
28C2×#14268626C2211yescharacteristic
29C2×#14268626C2211yescharacteristic
30C2×#14268624C2211yescharacteristic
31C2×#14268624C2211yescharacteristic
32C2×#14268624C2211yescharacteristic
33C2×#14268624C2211yescharacteristic
34C2×#14268624C2211yescharacteristic
35C2×#14268624C2211yescharacteristic
36C2×C4×Dic113C2211yescharacteristic
37C2×C4×Dic113C2211yescharacteristic
38C2×C4×Dic113C2211yescharacteristic
39C2×C4×Dic113C2211yescharacteristic
40C2×C4×Dic113C2211yescharacteristic
41#16813529C2211yescharacteristic
42#16813529C2211yescharacteristic
43#16813529C2211yescharacteristic
44#16813529C2211yescharacteristic
45#16813529C2211yescharacteristic
46#16813529C2211yescharacteristic
47C2×C4⋊C4×C113C2211yescharacteristic
48C2×C4⋊C4×C113C2211yescharacteristic
49C2×C42×C113C2211yescharacteristic
50#95×C113C2211yescharacteristic
51#95×C113C2211yescharacteristic
52#95×C113C2211yescharacteristic
53#95×C113C2211yescharacteristic
54C23×D113C2311yescharacteristic
55C2×Dic226—22no—
56C2×Dic226—22no—
57C2×Dic226C2×C414yes—
58C22×Dic113C2311yescharacteristic
59C22×Dic113C2311yescharacteristic
60C22×Dic113C2311yescharacteristic
61C22×Dic113C2311yescharacteristic
62C22×Dic113C2311yescharacteristic
63C22×Dic113C2311yescharacteristic
64C22×Dic113C2311yescharacteristic
65C2×C4×D113—22no—
66C2×C4×D113—22no—
67C2×C4×D113C2×C414yes—
68C4⋊Dic113—22no—
69C4⋊Dic113—22no—
70C4⋊Dic113—41no—
71#14268626—22no—
72#14268626—22no—
73#14268626—22no—
74#14268626—41no—
75#14268626C2×C414yes—
76#14268626C2×C414yes—
77#14268624—22no—
78#14268624—22no—
79#14268624—41no—
80#14268624—41no—
81#14268624C2×C414yes—
82#14268624C2×C414yes—
83C4×Dic113—22no—
84C4×Dic113—22no—
85C4×Dic113—41no—
86C4×Dic113—41no—
87C4×Dic113D414yes—
88C22×C4×C113C2311yescharacteristic
89C22×C4×C113C2311yescharacteristic
90C22×C4×C113C2311yescharacteristic
91C22×C4×C113C2311yescharacteristic
92C22×C4×C113C2311yescharacteristic
93C22×C4×C113C2311yescharacteristic
94C22×C4×C113C2311yescharacteristic
95C4⋊C4×C113—22no—
96C4⋊C4×C113C2×C414yes—
97C42×C113—22no—
98C23×C113C2411yescharacteristic
99C22×D113—21no—
100C22×D113—21no—
101C22×D113—21no—
102C22×D113—21no—
103C22×D113—21no—
104C22×D113—21no—
105C22×D113C22×C412yes—
106C2×C4×C113—21no—
107C2×C4×C113—21no—
108C2×C4×C113—21no—
109C2×C4×C113—21no—
110C2×C4×C113—21no—
111C2×C4×C113—21no—
112C2×C4×C113—21no—
113C2×C4×C113—21no—
114C2×C4×C113—21no—
115C2×C4×C113—21no—
116C2×C4×C113—21no—
117C2×C4×C113—21no—
118C2×C4×C113—21no—
119C2×C4×C113—21no—
120C2×C4×C113—21no—
121C2×C4×C113—21no—
122C2×C4×C113—21no—
123C2×C4×C113C22×C412yes—
124C2×C4×C113C22×C412yes—
125C2×C4×C113C22×C412yes—
126C2×C4×C113C2×D412yes—
127C2×Dic113—21no—
128C2×Dic113—21no—
129C2×Dic113—21no—
130C2×Dic113—21no—
131C2×Dic113—21no—
132C2×Dic113—21no—
133C2×Dic113—21no—
134C2×Dic113—21no—
135C2×Dic113—21no—
136C2×Dic113—21no—
137C2×Dic113—21no—
138C2×Dic113—21no—
139C2×Dic113—21no—
140C2×Dic113—21no—
141C2×Dic113—21no—
142C2×Dic113C22×C412yes—
143C2×Dic113C22×C412yes—
144C2×Dic113C22×C412yes—
145C2×Dic113C2×D412yes—
146C2×Dic113C2×D412yes—
147C2×Dic113#4112yes—
148C4×D113—28no—
149Dic226—28no—
150C22×C113C23×C411yescharacteristic
151C22×C113C22×D411yescharacteristic
152C22×C113C2×#4111yescharacteristic
153C22×C113H(5,𝔽2)11yescharacteristic
154C22×C1132−1+411yescharacteristic
155C22×C1132−1+411yescharacteristic
156C22×C1132−1+411yescharacteristic
157C2×D113—22no—
158C2×D113—22no—
159C2×D113—22no—
160C2×D113—41no—
161C2×D113—41no—
162C2×D113—41no—
163C2×D113—41no—
164C4×C113—22no—
165C4×C113—22no—
166C4×C113—22no—
167C4×C113—22no—
168C4×C113—41no—
169C4×C113—41no—
170C4×C113—41no—
171Dic113—22no—
172Dic113—22no—
173Dic113—22no—
174Dic113—41no—
175Dic113—41no—
176Dic113—41no—
177Dic113C4×D414yes—
178C2×C113C2×C4×D411yescharacteristic
179C2×C113#51911yescharacteristic
180C2×C113#52011yescharacteristic
181C2×C113#53611yescharacteristic
182C2×C113#53711yescharacteristic
183C2×C113#54111yescharacteristic
184C2×C113#54211yescharacteristic
185D113—42no—
186#2323—1131nomaximal, Sylow
187C113#232311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1131no—
189C2×C4×Q8—1131no—
190C2×#117—1131no—
191#391—1131no—
192#389—1131no—
193#389—1131no—
194#389—1131no—
195#388—1131no—
196#388—1131no—
197#388—1131no—
198#388—1131no—
199#387—1131no—
200#387D11311yescharacteristic, complemented
201#380—1131no—
202C4×C4⋊C4—1131no—
203C22×Q8—1131no—
204C23×C4—1131no—
205C22⋊Q8—1138no—
206C4×Q8—2264no—
207#117—2264no—
208C2×C4⋊C4—1131no—
209C2×C4⋊C4—1131no—
210C2×C4⋊C4—1131no—
211C2×C4⋊C4—1131no—
212C2×C4⋊C4—1131no—
213C2×C4⋊C4—1131no—
214C2×C4⋊C4—1131no—
215C2×C4⋊C4—1131no—
216C2×C4⋊C4—1131no—
217C2×C4⋊C4C2×D11311yescharacteristic
218C2×C4⋊C4C2×D11311yescharacteristic
219C2×C22⋊C4—1131no—
220C2×C22⋊C4—1131no—
221C2×C22⋊C4—1131no—
222C2×C22⋊C4—1131no—
223C2×C22⋊C4—1131no—
224C2×C22⋊C4—1131no—
225C2×C42—1131no—
226C2×C42—1131no—
227C2×C42—1131no—
228C2×C42—1131no—
229C2×C42—1131no—
230C2×C42C2×D11311yescharacteristic
231#95—1131no—
232#95—1131no—
233#95—1131no—
234#95—1131no—
235#95—1131no—
236#95—1131no—
237#95C2×D11311yescharacteristic
238#95C2×D11311yescharacteristic
239#95C2×D11311yescharacteristic
240#95C2×D11311yescharacteristic
241C24—1131no—
242C2×Q8—1134no—
243C2×Q8—2262no—
244C2×Q8—2262no—
245C22×C4—1131no—
246C22×C4—1131no—
247C22×C4—1131no—
248C22×C4—1131no—
249C22×C4—1134no—
250C22×C4—1131no—
251C22×C4—1131no—
252C22×C4—1131no—
253C22×C4—2262no—
254C22×C4—2262no—
255C22×C4C22×D11311yescharacteristic
256C22×C4C22×D11311yescharacteristic
257C22×C4C22×D11311yescharacteristic
258C22×C4C22×D11311yescharacteristic
259C22×C4C22×D11311yescharacteristic
260C22×C4C22×D11311yescharacteristic
261C22×C4C22×D11311yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—1134no—
264C4⋊C4—1134no—
265C4⋊C4—4521no—
266C4⋊C4—4521no—
267C4⋊C4—2262no—
268C4⋊C4—2262no—
269C4⋊C4—2262no—
270C4⋊C4—2262no—
271C4⋊C4—2262no—
272C4⋊C4C4×D11314yes—
273C22⋊C4—1134no—
274C22⋊C4—1134no—
275C22⋊C4—4521no—
276C22⋊C4—4521no—
277C22⋊C4—2262no—
278C22⋊C4—2262no—
279C42—22no—
280C42—1134no—
281C42—4521no—
282C42—4521no—
283C42—2262no—
284C42—2262no—
285C23—1132no—
286C23—2261no—
287C23—2261no—
288C23—2261no—
289C23—2261no—
290C23—2261no—
291C23—2261no—
292C23C23×D11311yescharacteristic
293Q8—2268no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1132no—
312C2×C4—1132no—
313C2×C4—1132no—
314C2×C4—1132no—
315C2×C4—1132no—
316C2×C4—1132no—
317C2×C4—2268no—
318C2×C4—2261no—
319C2×C4—2261no—
320C2×C4—2261no—
321C2×C4—2261no—
322C2×C4—2261no—
323C2×C4—2261no—
324C2×C4—2261no—
325C2×C4—2261no—
326C2×C4—2261no—
327C2×C4—2261no—
328C2×C4—2261no—
329C2×C4—2261no—
330C2×C4—2261no—
331C2×C4—2261no—
332C2×C4—2261no—
333C2×C4C2×C4×D11312yes—
334C2×C4C2×C4×D11312yes—
335C2×C4C2×C4×D11312yes—
336C2×C4D4×D11312yes—
337C22—4521no—
338C22—4521no—
339C22—4521no—
340C22—4521no—
341C22—2262no—
342C22—2262no—
343C22—2262no—
344C22C22×C4×D11311yescharacteristic
345C22C2×D4×D11311yescharacteristic
346C22#41×D11311yescharacteristic
347C22#1681366811yescharacteristic
348C22#1681366911yescharacteristic
349C22#1681366911yescharacteristic
350C22#1681367011yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—1134no—
359C4—4521no—
360C4—4521no—
361C4—4521no—
362C4—2262no—
363C4—2262no—
364C4—2262no—
365C2—4522no—
366C2C4×D4×D11311yescharacteristic
367C2#2029907411yescharacteristic
368C2#2029907811yescharacteristic
369C2#2029917011yescharacteristic
370C2#2029917811yescharacteristic
371C2#2029920711yescharacteristic
372C2#2029921411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D113
commutatorC22×C113C23×C4
Fitting#387×C113C2
socle, cosocleC23×C113C24
prime radicalC23×D113C23
hypercenter#387D113
hypocenterC113#2323
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic226C2semidirect
4. #20298502C2semidirect
5. #20298497C2semidirect
6. #20298495C2semidirect
7. #20298495C2semidirect
12. #20298479C2semidirect
14. #20298449C2semidirect
15. #387×C113C2semidirect
18. #16813600C4semidirect
187. C113#2323semidirect
200. #387D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136002
C2×C4×Dic2262
C2×#168135802
#202985022
#202984972
#202984952
#202984902
#202984892
#202984872
#202984792
#202984572
#202984492
#387×C1132
#2323113