Finite Group Databasebeta

← C23×#3120 #15670739

Subgroups — C23×#3120 #15670739

Order 2944.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#541C211yesmaximal, characteristic, complemented
2C23×#520C211yesmaximal, characteristic
3C23×#484C211yesmaximal, characteristic, complemented
4C23×#483C211yesmaximal, characteristic, complemented
5C23×#479C211yesmaximal, characteristic, complemented
6C23×#478C211yesmaximal, characteristic, complemented
7C23×#462C211yesmaximal, characteristic, complemented
8C23×#461C211yesmaximal, characteristic, complemented
9C23×#452C211yesmaximal, characteristic
10C23×#452C211yesmaximal, characteristic
11C23×#441C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C4×Q16×C23C211yesmaximal, characteristic, complemented
14C4×SD16×C23C211yesmaximal, characteristic, complemented
15C23×#433C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×Q16×C23C2211yescharacteristic
18C2×Q16×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×SD16×C23C2211yescharacteristic
21C23×C4⋊Q8C2211yescharacteristic
22C23×#126—21no—
23C23×#126—21no—
24C23×#125—21no—
25C23×#125—21no—
26C23×#125C2211yescharacteristic
27C23×C22⋊Q8C2211yescharacteristic
28C23×C22⋊Q8C2211yescharacteristic
29C4×Q8×C23C2211yescharacteristic
30C4×Q8×C23C2211yescharacteristic
31C4×Q8×C23C2211yescharacteristic
32C4×Q8×C23C2211yescharacteristic
33C4×D4×C23—21no—
34C4×D4×C23—21no—
35C4×D4×C23C2211yescharacteristic
36C4×D4×C23C2211yescharacteristic
37C23×#117—21no—
38C23×#117C2211yescharacteristic
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23C2211yescharacteristic
41C23×#107C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#106C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×C4⋊C8C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×#103C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×C22⋊C8C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×#97C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59#41×C23—22no—
60#41×C23—22no—
61C2×Q8×C23C2311yescharacteristic
62C2×Q8×C23C2311yescharacteristic
63C2×Q8×C23C2311yescharacteristic
64C2×D4×C23—21no—
65C2×D4×C23C2311yescharacteristic
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23C2311yescharacteristic
71Q16×C23—22no—
72Q16×C23—22no—
73SD16×C23—22no—
74SD16×C23—22no—
75C2×C8×C23C2311yescharacteristic
76C2×C8×C23C2311yescharacteristic
77C2×C8×C23C2311yescharacteristic
78C2×C8×C23C2311yescharacteristic
79C4⋊C4×C23—21no—
80C4⋊C4×C23—21no—
81C4⋊C4×C23—21no—
82C4⋊C4×C23—21no—
83C4⋊C4×C23—21no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23—21no—
86C4⋊C4×C23—21no—
87C4⋊C4×C23—21no—
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23D411yescharacteristic
91C4⋊C4×C23D411yescharacteristic
92C4⋊C4×C23C2311yescharacteristic
93C4⋊C4×C23C2311yescharacteristic
94C4⋊C4×C23C2311yescharacteristic
95C4⋊C4×C23C2311yescharacteristic
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C22⋊C4×C23—21no—
99C22⋊C4×C23—21no—
100C22⋊C4×C23D411yescharacteristic
101C22⋊C4×C23D411yescharacteristic
102C42×C23—21no—
103C42×C23—21no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23C2311yescharacteristic
107C42×C23C2311yescharacteristic
108C23×C23—21no—
109C23×C23C2×D411yescharacteristic
110Q8×C23—21no—
111Q8×C23—21no—
112Q8×C23—21no—
113Q8×C23—21no—
114Q8×C23—21no—
115Q8×C23#4112yes—
116D4×C23—24no—
117D4×C23—21no—
118D4×C23#4112yes—
119C2×C4×C23—24no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—81no—
130C2×C4×C23—81no—
131C2×C4×C23—81no—
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2×D411yescharacteristic
134C2×C4×C23C2×D411yescharacteristic
135C2×C4×C23C2×D411yescharacteristic
136C2×C4×C23C2×D411yescharacteristic
137C2×C4×C23C2411yescharacteristic
138C8×C23—21no—
139C8×C23—21no—
140C8×C23—21no—
141C8×C23—21no—
142#3120C2311yesmaximal, characteristic, direct product factor, Sylow
143C22×C23—21no—
144C22×C23—22no—
145C22×C23—21no—
146C22×C23—21no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23C22×D411yescharacteristic
150C4×C23—21no—
151C4×C23—22no—
152C4×C23—21no—
153C4×C23—21no—
154C4×C23—21no—
155C4×C23—21no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23C2×#4111yescharacteristic
163C4×C23C2×#4111yescharacteristic
164#541C2×C2311yescharacteristic, complemented
165#520C2×C2311yescharacteristic
166#484C2×C2311yescharacteristic, complemented
167#483C2×C2311yescharacteristic, complemented
168#479C2×C2311yescharacteristic, complemented
169#478C2×C2311yescharacteristic, complemented
170#462C2×C2311yescharacteristic, complemented
171#461C2×C2311yescharacteristic, complemented
172#452C2×C2311yescharacteristic
173#452C2×C2311yescharacteristic
174#441C2×C2311yescharacteristic, complemented
175#440C2×C2311yescharacteristic, complemented
176C4×Q16C2×C2311yescharacteristic, complemented
177C4×SD16C2×C2311yescharacteristic, complemented
178#433C2×C2311yescharacteristic, complemented
179C2×C23—41no—
180C2×C23—42no—
181C2×C23#52511yescharacteristic
182C2×C23#57711yescharacteristic
183C2×C23#57811yescharacteristic
184C2×#41C22×C2311yescharacteristic
185C2×Q16C22×C2311yescharacteristic
186C2×Q16C22×C2311yescharacteristic
187C2×SD16C22×C2311yescharacteristic
188C2×SD16C22×C2311yescharacteristic
189C4⋊Q8C22×C2311yescharacteristic
190#126—21no—
191#126—21no—
192#125—21no—
193#125—21no—
194#125C22×C2311yescharacteristic
195C22⋊Q8C22×C2311yescharacteristic
196C22⋊Q8C22×C2311yescharacteristic
197C4×Q8C22×C2311yescharacteristic
198C4×Q8C22×C2311yescharacteristic
199C4×Q8C22×C2311yescharacteristic
200C4×Q8C22×C2311yescharacteristic
201C4×D4—21no—
202C4×D4—21no—
203C4×D4C22×C2311yescharacteristic
204C4×D4C22×C2311yescharacteristic
205#117—21no—
206#117C22×C2311yescharacteristic
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C2311yescharacteristic
209#107C22×C2311yescharacteristic
210#107C22×C2311yescharacteristic
211#106C22×C2311yescharacteristic
212#106C22×C2311yescharacteristic
213C4⋊C8C22×C2311yescharacteristic
214C4⋊C8C22×C2311yescharacteristic
215#103C22×C2311yescharacteristic
216#103C22×C2311yescharacteristic
217#103C22×C2311yescharacteristic
218#103C22×C2311yescharacteristic
219#103C22×C2311yescharacteristic
220#103C22×C2311yescharacteristic
221#102C22×C2311yescharacteristic
222#102C22×C2311yescharacteristic
223C22⋊C8C22×C2311yescharacteristic
224C22⋊C8C22×C2311yescharacteristic
225#97C22×C2311yescharacteristic
226C4×C8C22×C2311yescharacteristic
227C23#312011yescharacteristic, direct product factor, Sylow
228#41—22no—
229#41—22no—
230C2×Q8C23×C2311yescharacteristic
231C2×Q8C23×C2311yescharacteristic
232C2×Q8C23×C2311yescharacteristic
233C2×D4—21no—
234C2×D4C23×C2311yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C2311yescharacteristic
240Q16—22no—
241Q16—22no—
242SD16—22no—
243SD16—22no—
244C2×C8C23×C2311yescharacteristic
245C2×C8C23×C2311yescharacteristic
246C2×C8C23×C2311yescharacteristic
247C2×C8C23×C2311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4D4×C2311yescharacteristic
260C4⋊C4D4×C2311yescharacteristic
261C4⋊C4C23×C2311yescharacteristic
262C4⋊C4C23×C2311yescharacteristic
263C4⋊C4C23×C2311yescharacteristic
264C4⋊C4C23×C2311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4D4×C2311yescharacteristic
270C22⋊C4D4×C2311yescharacteristic
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42C23×C2311yescharacteristic
276C42C23×C2311yescharacteristic
277C23—21no—
278C23C2×D4×C2311yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8#41×C2312yes—
285D4—24no—
286D4—21no—
287D4#41×C2312yes—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C2×D4×C2311yescharacteristic
302C2×C4C2×D4×C2311yescharacteristic
303C2×C4C2×D4×C2311yescharacteristic
304C2×C4C2×D4×C2311yescharacteristic
305C2×C4C2×D4×C2311yescharacteristic
306C2×C4C24×C2311yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22C22×D4×C2311yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C2×#41×C2311yescharacteristic
331C4C2×#41×C2311yescharacteristic
332C2—41no—
333C2—42no—
334C2C23×#52511yescharacteristic
335C2C23×#57711yescharacteristic
336C2C23×#57811yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC2×#41×C23C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#541C2semidirect
3. C23×#484C2semidirect
4. C23×#483C2semidirect
5. C23×#479C2semidirect
6. C23×#478C2semidirect
7. C23×#462C2semidirect
8. C23×#461C2semidirect
11. C23×#441C2semidirect
12. C23×#440C2semidirect
13. C4×Q16×C23C2semidirect
14. C4×SD16×C23C2semidirect
15. C23×#433C2semidirect
142. #3120C23direct (N × H)
164. #541C2×C23semidirect
166. #484C2×C23semidirect
167. #483C2×C23semidirect
168. #479C2×C23semidirect
169. #478C2×C23semidirect
170. #462C2×C23semidirect
171. #461C2×C23semidirect
174. #441C2×C23semidirect
175. #440C2×C23semidirect
176. C4×Q16C2×C23semidirect
177. C4×SD16C2×C23semidirect
178. #433C2×C23semidirect
227. C23#3120direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5412
C23×#5202
C23×#4842
C23×#4832
C23×#4792
C23×#4782
C23×#4622
C23×#4612
C23×#4522
C23×#4412
C23×#4402
C4×Q16×C232
C4×SD16×C232
C23×#4332
#312023