Finite Group Databasebeta

← C23×#3370 #15670989

Subgroups — C23×#3370 #15670989

Order 2944.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#559C211yesmaximal, characteristic, complemented
2C23×#554C211yesmaximal, characteristic, complemented
3C23×#490C211yesmaximal, characteristic, complemented
4C23×#489C211yesmaximal, characteristic, complemented
5C23×#488C211yesmaximal, characteristic, complemented
6C23×#479C211yesmaximal, characteristic, complemented
7C23×#478C211yesmaximal, characteristic, complemented
8C23×#474C211yesmaximal, characteristic, complemented
9C23×#463C211yesmaximal, characteristic, complemented
10C23×#463C211yesmaximal, characteristic, complemented
11C23×#459C211yesmaximal, characteristic
12C23×#446C211yesmaximal, characteristic, complemented
13C23×#442C211yesmaximal, characteristic, complemented
14C23×#440C211yesmaximal, characteristic, complemented
15C4×D8×C23C211yesmaximal, characteristic, complemented
16C2×SD16×C23C2211yescharacteristic
17C2×D8×C23C2211yescharacteristic
18C2×D8×C23C2211yescharacteristic
19C23×C4⋊Q8C2211yescharacteristic
20C23×#127C2211yescharacteristic
21C23×#125C2211yescharacteristic
22C23×#125C2211yescharacteristic
23C23×#124—21no—
24C23×#124C2211yescharacteristic
25C23×#124C2211yescharacteristic
26C23×#123—21no—
27C23×#123—21no—
28C23×C22⋊Q8—21no—
29C23×C22⋊Q8—21no—
30C23×C22⋊Q8—21no—
31C4×Q8×C23C2211yescharacteristic
32C4×Q8×C23C2211yescharacteristic
33C4×D4×C23—21no—
34C4×D4×C23C2211yescharacteristic
35C4×D4×C23C2211yescharacteristic
36C4×D4×C23C2211yescharacteristic
37C4×D4×C23C2211yescharacteristic
38C4×D4×C23C2211yescharacteristic
39C2×C4⋊C4×C23—21no—
40C23×#107C2211yescharacteristic
41C23×#107C2211yescharacteristic
42C23×#106C2211yescharacteristic
43C23×C4⋊C8C2211yescharacteristic
44C23×C4⋊C8C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×#103C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×#97C2211yescharacteristic
57C23×#97C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59C2×Q8×C23—21no—
60C2×Q8×C23D411yescharacteristic
61C2×Q8×C23C2311yescharacteristic
62C2×D4×C23—21no—
63C2×D4×C23—21no—
64C2×D4×C23C2311yescharacteristic
65C2×D4×C23C2311yescharacteristic
66C2×D4×C23C2311yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72SD16×C23—22no—
73D8×C23—22no—
74D8×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×C23D411yescharacteristic
86C4⋊C4×C23D411yescharacteristic
87C4⋊C4×C23D411yescharacteristic
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2311yescharacteristic
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C22⋊C4×C23—21no—
93C22⋊C4×C23—21no—
94C22⋊C4×C23—21no—
95C22⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C22⋊C4×C23—21no—
99C22⋊C4×C23—21no—
100C22⋊C4×C23—21no—
101C42×C23—21no—
102C42×C23C2311yescharacteristic
103C42×C23C2311yescharacteristic
104C42×C23C2311yescharacteristic
105C23×C23—21no—
106C23×C23—21no—
107C23×C23—21no—
108Q8×C23—21no—
109Q8×C23—21no—
110Q8×C23—41no—
111Q8×C23—81no—
112D4×C23—21no—
113D4×C23—21no—
114D4×C23—21no—
115D4×C23—21no—
116D4×C23—21no—
117D4×C23—42no—
118D4×C23—81no—
119D4×C23#4112yes—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—42no—
126C2×C4×C23—81no—
127C2×C4×C23—81no—
128C2×C4×C23—81no—
129C2×C4×C23—81no—
130C2×C4×C23C2×D411yescharacteristic
131C2×C4×C23C2×D411yescharacteristic
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2×D411yescharacteristic
134C2×C4×C23C2×D411yescharacteristic
135C2×C4×C23C2×D411yescharacteristic
136C2×C4×C23C2411yescharacteristic
137C8×C23—21no—
138C8×C23—21no—
139C8×C23—21no—
140C8×C23—21no—
141#3370C2311yesmaximal, characteristic, direct product factor, Sylow
142C22×C23—22no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23—41no—
151C22×C23C22×D411yescharacteristic
152C4×C23—21no—
153C4×C23—21no—
154C4×C23—21no—
155C4×C23—21no—
156C4×C23—21no—
157C4×C23—21no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—41no—
163C4×C23C2×#4111yescharacteristic
164C4×C232−1+411yescharacteristic
165#559C2×C2311yescharacteristic, complemented
166#554C2×C2311yescharacteristic, complemented
167#490C2×C2311yescharacteristic, complemented
168#489C2×C2311yescharacteristic, complemented
169#488C2×C2311yescharacteristic, complemented
170#479C2×C2311yescharacteristic, complemented
171#478C2×C2311yescharacteristic, complemented
172#474C2×C2311yescharacteristic, complemented
173#463C2×C2311yescharacteristic, complemented
174#463C2×C2311yescharacteristic, complemented
175#459C2×C2311yescharacteristic
176#446C2×C2311yescharacteristic, complemented
177#442C2×C2311yescharacteristic, complemented
178#440C2×C2311yescharacteristic, complemented
179C4×D8C2×C2311yescharacteristic, complemented
180C2×C23—42no—
181C2×C23—81no—
182C2×C23—81no—
183C2×C23#54811yescharacteristic
184C2×C23#57511yescharacteristic
185C2×C23#57611yescharacteristic
186C2×SD16C22×C2311yescharacteristic
187C2×D8C22×C2311yescharacteristic
188C2×D8C22×C2311yescharacteristic
189C4⋊Q8C22×C2311yescharacteristic
190#127C22×C2311yescharacteristic
191#125C22×C2311yescharacteristic
192#125C22×C2311yescharacteristic
193#124—21no—
194#124C22×C2311yescharacteristic
195#124C22×C2311yescharacteristic
196#123—21no—
197#123—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C4×Q8C22×C2311yescharacteristic
202C4×Q8C22×C2311yescharacteristic
203C4×D4—21no—
204C4×D4C22×C2311yescharacteristic
205C4×D4C22×C2311yescharacteristic
206C4×D4C22×C2311yescharacteristic
207C4×D4C22×C2311yescharacteristic
208C4×D4C22×C2311yescharacteristic
209C2×C4⋊C4—21no—
210#107C22×C2311yescharacteristic
211#107C22×C2311yescharacteristic
212#106C22×C2311yescharacteristic
213C4⋊C8C22×C2311yescharacteristic
214C4⋊C8C22×C2311yescharacteristic
215C4⋊C8C22×C2311yescharacteristic
216#103C22×C2311yescharacteristic
217#103C22×C2311yescharacteristic
218#103C22×C2311yescharacteristic
219#102C22×C2311yescharacteristic
220#102C22×C2311yescharacteristic
221#102C22×C2311yescharacteristic
222#102C22×C2311yescharacteristic
223#102C22×C2311yescharacteristic
224#102C22×C2311yescharacteristic
225#102C22×C2311yescharacteristic
226#97C22×C2311yescharacteristic
227#97C22×C2311yescharacteristic
228C4×C8C22×C2311yescharacteristic
229C23#337011yescharacteristic, direct product factor, Sylow
230C2×Q8—21no—
231C2×Q8D4×C2311yescharacteristic
232C2×Q8C23×C2311yescharacteristic
233C2×D4—21no—
234C2×D4—21no—
235C2×D4C23×C2311yescharacteristic
236C2×D4C23×C2311yescharacteristic
237C2×D4C23×C2311yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243SD16—22no—
244D8—22no—
245D8—22no—
246C2×C8C23×C2311yescharacteristic
247C2×C8C23×C2311yescharacteristic
248C2×C8C23×C2311yescharacteristic
249C2×C8C23×C2311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4D4×C2311yescharacteristic
257C4⋊C4D4×C2311yescharacteristic
258C4⋊C4D4×C2311yescharacteristic
259C4⋊C4C23×C2311yescharacteristic
260C4⋊C4C23×C2311yescharacteristic
261C4⋊C4C23×C2311yescharacteristic
262C4⋊C4C23×C2311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C42—21no—
273C42C23×C2311yescharacteristic
274C42C23×C2311yescharacteristic
275C42C23×C2311yescharacteristic
276C23—21no—
277C23—21no—
278C23—21no—
279Q8—21no—
280Q8—21no—
281Q8—41no—
282Q8—81no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—42no—
289D4—81no—
290D4#41×C2312yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—81no—
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×C4C2×D4×C2311yescharacteristic
307C2×C4C24×C2311yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C22×D4×C2311yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4C2×#41×C2311yescharacteristic
334C4C23×2−1+411yescharacteristic
335C2—42no—
336C2—81no—
337C2—81no—
338C2C23×#54811yescharacteristic
339C2C23×#57511yescharacteristic
340C2C23×#57611yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#459C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#559C2semidirect
2. C23×#554C2semidirect
3. C23×#490C2semidirect
4. C23×#489C2semidirect
5. C23×#488C2semidirect
6. C23×#479C2semidirect
7. C23×#478C2semidirect
8. C23×#474C2semidirect
9. C23×#463C2semidirect
10. C23×#463C2semidirect
12. C23×#446C2semidirect
13. C23×#442C2semidirect
14. C23×#440C2semidirect
15. C4×D8×C23C2semidirect
141. #3370C23direct (N × H)
165. #559C2×C23semidirect
166. #554C2×C23semidirect
167. #490C2×C23semidirect
168. #489C2×C23semidirect
169. #488C2×C23semidirect
170. #479C2×C23semidirect
171. #478C2×C23semidirect
172. #474C2×C23semidirect
173. #463C2×C23semidirect
174. #463C2×C23semidirect
176. #446C2×C23semidirect
177. #442C2×C23semidirect
178. #440C2×C23semidirect
179. C4×D8C2×C23semidirect
229. C23#3370direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5592
C23×#5542
C23×#4902
C23×#4892
C23×#4882
C23×#4792
C23×#4782
C23×#4742
C23×#4632
C23×#4592
C23×#4462
C23×#4422
C23×#4402
C4×D8×C232
#337023