Finite Group Databasebeta

← C4⋊C4×#41×C103 #39939389

Subgroups — C4⋊C4×#41×C103 #39939389

Order 26368.

3250 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41×C103C212yesmaximal
2C2×C4×#41×C103C211yesmaximal, characteristic
3C103×#2351C212yesmaximal, complemented
4Q8×C4⋊C4×C103C212yesmaximal, complemented
5C103×#2349C216yesmaximal, complemented
6D4×C4⋊C4×C103C216yesmaximal, complemented
7C103×#2296C216yesmaximal, complemented
8C103×#2285C213yesmaximal, complemented
9C2×C4×C4⋊C4×C103C213yesmaximal, complemented
10C22×#41×C103C2211yescharacteristic
11C4×#41×C103—216no—
12C4×#41×C103C4116yescomplemented
13C2×C4×Q8×C103C2214yes—
14C2×C4×Q8×C103C2211yescharacteristic
15C2×C4×Q8×C103C2211yescharacteristic
16C2×C4×D4×C103C22112yes—
17C2×C4×D4×C103C2213yes—
18C2×C4×D4×C103C2213yes—
19C2×#117×C103C2216yes—
20C2×#117×C103C2213yes—
21C22×C4⋊C4×C103C2216yes—
22C22×C42×C103C2216yes—
23C22×C42×C103C2213yes—
24#389×C103C22112yes—
25#389×C103C22112yes—
26#385×C103C22124yes—
27#384×C103C2216yes—
28#382×C103C2216yes—
29#380×C103C2216yes—
30#380×C103C2216yes—
31C4×C4⋊C4×C103C2211yescharacteristic
32C4×C4⋊C4×C103C2213yes—
33C4×C4⋊C4×C103C2216yes—
34C4×C4⋊C4×C103C2216yes—
35C4×C4⋊C4×C103C2216yes—
36C4×C4⋊C4×C103C2216yes—
37C2×#95×C103C2216yes—
38C2×#41×C103C2×C418yes—
39C2×#41×C103D418yes—
40C2×#41×C103Q818yes—
41C22×Q8×C103C2311yescharacteristic
42C22×D4×C103C2313yes—
43C23×C4×C103C23112yes—
44C23×C4×C103C2313yes—
45C23×C4×C103C2313yes—
46C23×C4×C103C2313yes—
47C4×Q8×C103—216no—
48C4×Q8×C103C2×C418yescomplemented
49C4×Q8×C103C2×C418yescomplemented
50C4×D4×C103—248no—
51C4×D4×C103C2×C4124yescomplemented
52C4×D4×C103C2×C4124yescomplemented
53#117×C103—224no—
54#117×C103C2×C4124yescomplemented
55C2×C4⋊C4×C103—224no—
56C2×C4⋊C4×C103C2316yes—
57C2×C4⋊C4×C103C23112yes—
58C2×C4⋊C4×C103C23112yes—
59C2×C4⋊C4×C103C23112yes—
60C2×C4⋊C4×C103C2313yes—
61C2×C4⋊C4×C103C2313yes—
62C2×C4⋊C4×C103C2312yes—
63C2×C22⋊C4×C103C23112yes—
64C2×C22⋊C4×C103C2316yes—
65C2×C42×C103—224no—
66C2×C42×C103C2×C4124yescomplemented
67C2×C42×C103C23112yes—
68C2×C42×C103C2312yes—
69C2×C42×C103C2316yes—
70C2×C42×C103C2313yes—
71C2×C42×C103C2313yes—
72C2×C42×C103C2311yescharacteristic
73C2×C42×C103C2313yes—
74#95×C103C2312yes—
75#95×C103C2316yes—
76#95×C103C23112yes—
77#95×C103C23112yes—
78C24×C103C2413yes—
79#41×C103C4⋊C4164yesdirect product factor
80C2×Q8×C103C22×C414yes—
81C2×Q8×C103C2×D414yes—
82C2×Q8×C103C2×Q814yes—
83C2×D4×C103C22×C4112yes—
84C2×D4×C103C2×D4112yes—
85C2×D4×C103C2×Q8112yes—
86C22×C4×C103—212no—
87C22×C4×C103—224no—
88C22×C4×C103—224no—
89C22×C4×C103—46no—
90C22×C4×C103—412no—
91C22×C4×C103C22×C4112yes—
92C22×C4×C103C22×C4112yes—
93C22×C4×C103C22×C4112yes—
94C22×C4×C103C2×D4112yes—
95C22×C4×C103C2×Q8112yes—
96C22×C4×C103C24112yes—
97C22×C4×C103C2411yescharacteristic
98C22×C4×C103C2413yes—
99C22×C4×C103C2413yes—
100C22×C4×C103C2414yes—
101C22×C4×C103C2413yes—
102C22×C4×C103C2411yescharacteristic
103C22×C4×C103C2411yescharacteristic
104C4⋊C4×C103—224no—
105C4⋊C4×C103—224no—
106C4⋊C4×C103—212no—
107C4⋊C4×C103—412no—
108C4⋊C4×C103C22×C4112yes—
109C4⋊C4×C103C22×C4112yes—
110C4⋊C4×C103#4118yesdirect product factor
111C22⋊C4×C103—224no—
112C22⋊C4×C103C22×C4124yes—
113C42×C103—24no—
114C42×C103—212no—
115C42×C103—224no—
116C42×C103C22×C414yes—
117C42×C103C22×C4112yes—
118C42×C103C22×C4112yes—
119C42×C103C22×C4112yes—
120C23×C103—26no—
121C23×C103—26no—
122C23×C103C23×C416yes—
123C23×C103C22×D416yes—
124C23×C103C22×Q816yes—
125C23×C103C2511yescharacteristic
126Q8×C103C2×C4⋊C4116yescomplemented
127D4×C103C2×C4⋊C4148yescomplemented
128C2×C4×C103—24no—
129C2×C4×C103—26no—
130C2×C4×C103—26no—
131C2×C4×C103—22no—
132C2×C4×C103—24no—
133C2×C4×C103—224no—
134C2×C4×C103—212no—
135C2×C4×C103—212no—
136C2×C4×C103—212no—
137C2×C4×C103—412no—
138C2×C4×C103—448no—
139C2×C4×C103C2×C4⋊C4148yescomplemented
140C2×C4×C103C23×C412yes—
141C2×C4×C103C23×C416yes—
142C2×C4×C103C23×C416yes—
143C2×C4×C103C23×C412yes—
144C2×C4×C103C23×C416yes—
145C2×C4×C103C23×C412yes—
146C2×C4×C103C22×D412yes—
147C2×C4×C103C22×D416yes—
148C2×C4×C103C22×Q812yes—
149C2×C4×C103C22×Q816yes—
150C2×C4×C103C2×#4118yes—
151C2×C4×C103C2×#4114yes—
152C22×C103—26no—
153C22×C103—26no—
154C22×C103—212no—
155C22×C103—212no—
156C22×C103C22×C4⋊C4112yes—
157C22×C103C24×C411yescharacteristic
158C22×C103C23×D411yescharacteristic
159C22×C103C23×Q811yescharacteristic
160C22×C103C22×#4112yes—
161C22×C103#58512yes—
162C4×C103—26no—
163C4×C103—22no—
164C4×C103—26no—
165C4×C103—28no—
166C4×C103—412no—
167C4×C103C22×C4⋊C414yes—
168C4×C103C22×C4⋊C4112yes—
169C4×C103C4×#4114yescomplemented
170C4⋊C4×#41C10311yesmaximal, characteristic, direct product factor, Sylow
171C2×C103—212no—
172C2×C103C23×C4⋊C411yescharacteristic
173C2×C103C2×C4×#4111yescharacteristic
174C2×C103#342811yescharacteristic
175C2×C103D4×#4112yes—
176C2×C103Q8×#4112yes—
177C2×C4×#41C2×C10312yes—
178C2×C4×#41C2×C10311yescharacteristic
179#2351C2×C10312yescomplemented
180Q8×C4⋊C4C2×C10312yescomplemented
181#2349C2×C10316yescomplemented
182D4×C4⋊C4C2×C10316yescomplemented
183#2296C2×C10316yescomplemented
184#2285C2×C10313yescomplemented
185C2×C4×C4⋊C4C2×C10313yescomplemented
186C103C4⋊C4×#4111yescharacteristic, direct product factor, Sylow
187C22×#41C22×C10311yescharacteristic
188C4×#41—216no—
189C4×#41C4×C103116yescomplemented
190C2×C4×Q8C22×C10314yes—
191C2×C4×Q8C22×C10311yescharacteristic
192C2×C4×Q8C22×C10311yescharacteristic
193C2×C4×D4C22×C103112yes—
194C2×C4×D4C22×C10313yes—
195C2×C4×D4C22×C10313yes—
196C2×#117C22×C10316yes—
197C2×#117C22×C10313yes—
198C22×C4⋊C4C22×C10316yes—
199C22×C42C22×C10316yes—
200C22×C42C22×C10313yes—
201#389C22×C103112yes—
202#389C22×C103112yes—
203#385C22×C103124yes—
204#384C22×C10316yes—
205#382C22×C10316yes—
206#380C22×C10316yes—
207#380C22×C10316yes—
208C4×C4⋊C4C22×C10316yes—
209C4×C4⋊C4C22×C10316yes—
210C4×C4⋊C4C22×C10311yescharacteristic
211C4×C4⋊C4C22×C10313yes—
212C4×C4⋊C4C22×C10316yes—
213C4×C4⋊C4C22×C10316yes—
214C2×#95C22×C10316yes—
215C2×#41C2×C4×C10318yes—
216C2×#41D4×C10318yes—
217C2×#41Q8×C10318yes—
218C22×Q8C23×C10311yescharacteristic
219C22×D4C23×C10313yes—
220C23×C4C23×C103112yes—
221C23×C4C23×C10313yes—
222C23×C4C23×C10313yes—
223C23×C4C23×C10313yes—
224C4×Q8—216no—
225C4×Q8C2×C4×C10318yescomplemented
226C4×Q8C2×C4×C10318yescomplemented
227C4×D4—248no—
228C4×D4C2×C4×C103124yescomplemented
229C4×D4C2×C4×C103124yescomplemented
230#117—224no—
231#117C2×C4×C103124yescomplemented
232C2×C4⋊C4—224no—
233C2×C4⋊C4C23×C10316yes—
234C2×C4⋊C4C23×C103112yes—
235C2×C4⋊C4C23×C103112yes—
236C2×C4⋊C4C23×C103112yes—
237C2×C4⋊C4C23×C10313yes—
238C2×C4⋊C4C23×C10313yes—
239C2×C4⋊C4C23×C10312yes—
240C2×C22⋊C4C23×C103112yes—
241C2×C22⋊C4C23×C10316yes—
242C2×C42—224no—
243C2×C42C2×C4×C103124yescomplemented
244C2×C42C23×C10316yes—
245C2×C42C23×C103112yes—
246C2×C42C23×C10312yes—
247C2×C42C23×C10313yes—
248C2×C42C23×C10313yes—
249C2×C42C23×C10313yes—
250C2×C42C23×C10311yescharacteristic
251#95C23×C10312yes—
252#95C23×C10316yes—
253#95C23×C103112yes—
254#95C23×C103112yes—
255C24C24×C10313yes—
256#41C4⋊C4×C103164yesdirect product factor
257C2×Q8C22×C4×C10314yes—
258C2×Q8C2×D4×C10314yes—
259C2×Q8C2×Q8×C10314yes—
260C2×D4C22×C4×C103112yes—
261C2×D4C2×D4×C103112yes—
262C2×D4C2×Q8×C103112yes—
263C22×C4—212no—
264C22×C4—224no—
265C22×C4—224no—
266C22×C4—46no—
267C22×C4—412no—
268C22×C4C22×C4×C103112yes—
269C22×C4C22×C4×C103112yes—
270C22×C4C22×C4×C103112yes—
271C22×C4C2×D4×C103112yes—
272C22×C4C2×Q8×C103112yes—
273C22×C4C24×C103112yes—
274C22×C4C24×C10311yescharacteristic
275C22×C4C24×C10313yes—
276C22×C4C24×C10313yes—
277C22×C4C24×C10314yes—
278C22×C4C24×C10313yes—
279C22×C4C24×C10311yescharacteristic
280C22×C4C24×C10311yescharacteristic
281C4⋊C4—224no—
282C4⋊C4—224no—
283C4⋊C4—212no—
284C4⋊C4—412no—
285C4⋊C4C22×C4×C103112yes—
286C4⋊C4C22×C4×C103112yes—
287C4⋊C4#41×C10318yesdirect product factor
288C22⋊C4—224no—
289C22⋊C4C22×C4×C103124yes—
290C42—24no—
291C42—212no—
292C42—224no—
293C42C22×C4×C10314yes—
294C42C22×C4×C103112yes—
295C42C22×C4×C103112yes—
296C42C22×C4×C103112yes—
297C23—26no—
298C23—26no—
299C23C23×C4×C10316yes—
300C23C22×D4×C10316yes—
301C23C22×Q8×C10316yes—
302C23C25×C10311yescharacteristic
303Q8C2×C4⋊C4×C103116yescomplemented
304D4C2×C4⋊C4×C103148yescomplemented
305C2×C4—24no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—224no—
311C2×C4—212no—
312C2×C4—212no—
313C2×C4—212no—
314C2×C4—412no—
315C2×C4—448no—
316C2×C4C2×C4⋊C4×C103148yescomplemented
317C2×C4C23×C4×C10312yes—
318C2×C4C23×C4×C10316yes—
319C2×C4C23×C4×C10316yes—
320C2×C4C23×C4×C10312yes—
321C2×C4C23×C4×C10316yes—
322C2×C4C23×C4×C10312yes—
323C2×C4C22×D4×C10312yes—
324C2×C4C22×D4×C10316yes—
325C2×C4C22×Q8×C10312yes—
326C2×C4C22×Q8×C10316yes—
327C2×C4C2×#41×C10318yes—
328C2×C4C2×#41×C10314yes—
329C22—212no—
330C22—212no—
331C22—26no—
332C22—26no—
333C22C22×C4⋊C4×C103112yes—
334C22C24×C4×C10311yescharacteristic
335C22C23×D4×C10311yescharacteristic
336C22C23×Q8×C10311yescharacteristic
337C22C22×#41×C10312yes—
338C22#585×C10312yes—
339C4—28no—
340C4—26no—
341C4—22no—
342C4—26no—
343C4—412no—
344C4C22×C4⋊C4×C10314yes—
345C4C22×C4⋊C4×C103112yes—
346C4C4×#41×C10314yescomplemented
347C2—212no—
348C2C23×C4⋊C4×C10311yescharacteristic
349C2C2×C4×#41×C10311yescharacteristic
350C2C103×#342811yescharacteristic
351C2D4×#41×C10312yes—
352C2Q8×#41×C10312yes—
subgroupG/H
centerC22×C4×C103C24
commutatorC22C24×C4×C103
Frattini, cosocleC23C25×C103
socleC23×C103C25
prime radicalC22×#41×C103C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C103 ◁ G

Complements

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

NH ≅ G/Ntype
3. C103×#2351C2semidirect
4. Q8×C4⋊C4×C103C2semidirect
5. C103×#2349C2semidirect
6. D4×C4⋊C4×C103C2semidirect
7. C103×#2296C2semidirect
8. C103×#2285C2semidirect
9. C2×C4×C4⋊C4×C103C2semidirect
12. C4×#41×C103C4semidirect
48. C4×Q8×C103C2×C4semidirect
49. C4×Q8×C103C2×C4semidirect
51. C4×D4×C103C2×C4semidirect
52. C4×D4×C103C2×C4semidirect
54. #117×C103C2×C4semidirect
66. C2×C42×C103C2×C4semidirect
79. #41×C103C4⋊C4direct (N × H)
110. C4⋊C4×C103#41direct (N × H)
126. Q8×C103C2×C4⋊C4semidirect
127. D4×C103C2×C4⋊C4semidirect
139. C2×C4×C103C2×C4⋊C4semidirect
169. C4×C103C4×#41semidirect
170. C4⋊C4×#41C103direct (N × H)
179. #2351C2×C103semidirect
180. Q8×C4⋊C4C2×C103semidirect
181. #2349C2×C103semidirect
182. D4×C4⋊C4C2×C103semidirect
183. #2296C2×C103semidirect
184. #2285C2×C103semidirect
185. C2×C4×C4⋊C4C2×C103semidirect
186. C103C4⋊C4×#41direct (N × H)
189. C4×#41C4×C103semidirect
225. C4×Q8C2×C4×C103semidirect
226. C4×Q8C2×C4×C103semidirect
228. C4×D4C2×C4×C103semidirect
229. C4×D4C2×C4×C103semidirect
231. #117C2×C4×C103semidirect
243. C2×C42C2×C4×C103semidirect
256. #41C4⋊C4×C103direct (N × H)
287. C4⋊C4#41×C103direct (N × H)
303. Q8C2×C4⋊C4×C103semidirect
304. D4C2×C4⋊C4×C103semidirect
316. C2×C4C2×C4⋊C4×C103semidirect
346. C4C4×#41×C103semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#41×C1032
C103×#23512
Q8×C4⋊C4×C1032
C103×#23492
D4×C4⋊C4×C1032
C103×#22962
C103×#22852
C2×C4×C4⋊C4×C1032
C4⋊C4×#41103