Finite Group Databasebeta

← C23×#3347 #15670966

Subgroups — C23×#3347 #15670966

Order 2944.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#555C211yesmaximal, characteristic, complemented
2C23×#553C211yesmaximal, characteristic, complemented
3C23×#501C211yesmaximal, characteristic, complemented
4C23×#482C211yesmaximal, characteristic, complemented
5C23×#482C211yesmaximal, characteristic, complemented
6C23×#482C211yesmaximal, characteristic, complemented
7C23×#480C211yesmaximal, characteristic, complemented
8C23×#478C211yesmaximal, characteristic, complemented
9C23×#474C211yesmaximal, characteristic, complemented
10C23×#469C211yesmaximal, characteristic, complemented
11C23×#466C211yesmaximal, characteristic, complemented
12C23×#442C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#428C211yesmaximal, characteristic, complemented
15C23×#427C211yesmaximal, characteristic, complemented
16C2×D8×C23C2211yescharacteristic
17C2×M16×C23C2211yescharacteristic
18C22×C8×C23C2211yescharacteristic
19C23×C4⋊Q8C2211yescharacteristic
20C23×#126—21no—
21C23×#125C2211yescharacteristic
22C23×#124—21no—
23C23×#124—21no—
24C23×#123—21no—
25C23×#121—21no—
26C23×#121—21no—
27C23×#121C2211yescharacteristic
28C23×#121C2211yescharacteristic
29C23×#121C2211yescharacteristic
30C23×#121C2211yescharacteristic
31C4×D4×C23—21no—
32C4×D4×C23C2211yescharacteristic
33C4×D4×C23C2211yescharacteristic
34C4×D4×C23C2211yescharacteristic
35C23×#117—21no—
36C23×#117C2211yescharacteristic
37C23×#117C2211yescharacteristic
38C23×#117C2211yescharacteristic
39C2×C4⋊C4×C23C2211yescharacteristic
40C23×#107C2211yescharacteristic
41C23×#107C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#107C2211yescharacteristic
44C23×#107C2211yescharacteristic
45C23×#106C2211yescharacteristic
46C23×#106C2211yescharacteristic
47C23×#106C2211yescharacteristic
48C23×#106C2211yescharacteristic
49C23×C4⋊C8C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C23×#97C2211yescharacteristic
59C2×Q8×C23—21no—
60C2×D4×C23—21no—
61C2×D4×C23—21no—
62C2×D4×C23—21no—
63C2×D4×C23—21no—
64C2×D4×C23D411yescharacteristic
65C2×D4×C23C2311yescharacteristic
66C2×D4×C23C2311yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23C2311yescharacteristic
71C22×C4×C23C2311yescharacteristic
72D8×C23—22no—
73M16×C23—22no—
74C2×C8×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×C23D411yescharacteristic
84C4⋊C4×C23C2311yescharacteristic
85C4⋊C4×C23C2311yescharacteristic
86C4⋊C4×C23C2311yescharacteristic
87C4⋊C4×C23C2311yescharacteristic
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2311yescharacteristic
90C22⋊C4×C23—21no—
91C22⋊C4×C23—21no—
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×C23D411yescharacteristic
100C22⋊C4×C23D411yescharacteristic
101C42×C23—21no—
102C42×C23—21no—
103C42×C23—21no—
104C42×C23C2311yescharacteristic
105C23×C23—21no—
106C23×C23—21no—
107C23×C23C2×D411yescharacteristic
108C23×C23C2×D411yescharacteristic
109Q8×C23—81no—
110D4×C23—21no—
111D4×C23—21no—
112D4×C23—21no—
113D4×C23—21no—
114D4×C23—41no—
115D4×C23—42no—
116D4×C23—81no—
117D4×C23—81no—
118D4×C23—81no—
119C2×C4×C23—21no—
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—42no—
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×C23C2411yescharacteristic
137C8×C23—21no—
138C8×C23—21no—
139C8×C23—21no—
140C8×C23#4112yes—
141#3347C2311yesmaximal, characteristic, direct product factor, Sylow
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—21no—
146C22×C23—21no—
147C22×C23—21no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23—41no—
151C22×C23—41no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23C22×D411yescharacteristic
155C4×C23—21no—
156C4×C23—21no—
157C4×C23—21no—
158C4×C23—21no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—41no—
163C4×C23—41no—
164C4×C23—41no—
165C4×C23C2×#4111yescharacteristic
166C4×C232−1+411yescharacteristic
167#555C2×C2311yescharacteristic, complemented
168#553C2×C2311yescharacteristic, complemented
169#501C2×C2311yescharacteristic, complemented
170#482C2×C2311yescharacteristic, complemented
171#482C2×C2311yescharacteristic, complemented
172#482C2×C2311yescharacteristic, complemented
173#480C2×C2311yescharacteristic, complemented
174#478C2×C2311yescharacteristic, complemented
175#474C2×C2311yescharacteristic, complemented
176#469C2×C2311yescharacteristic, complemented
177#466C2×C2311yescharacteristic, complemented
178#442C2×C2311yescharacteristic, complemented
179#435C2×C2311yescharacteristic, complemented
180#428C2×C2311yescharacteristic, complemented
181#427C2×C2311yescharacteristic, complemented
182C2×C23—41no—
183C2×C23—41no—
184C2×C23—81no—
185C2×C23—81no—
186C2×C23#54711yescharacteristic
187C2×C23#57511yescharacteristic
188C2×C23#57611yescharacteristic
189C2×D8C22×C2311yescharacteristic
190C2×M16C22×C2311yescharacteristic
191C22×C8C22×C2311yescharacteristic
192C4⋊Q8C22×C2311yescharacteristic
193#126—21no—
194#125C22×C2311yescharacteristic
195#124—21no—
196#124—21no—
197#123—21no—
198#121—21no—
199#121—21no—
200#121C22×C2311yescharacteristic
201#121C22×C2311yescharacteristic
202#121C22×C2311yescharacteristic
203#121C22×C2311yescharacteristic
204C4×D4—21no—
205C4×D4C22×C2311yescharacteristic
206C4×D4C22×C2311yescharacteristic
207C4×D4C22×C2311yescharacteristic
208#117—21no—
209#117C22×C2311yescharacteristic
210#117C22×C2311yescharacteristic
211#117C22×C2311yescharacteristic
212C2×C4⋊C4C22×C2311yescharacteristic
213#107C22×C2311yescharacteristic
214#107C22×C2311yescharacteristic
215#107C22×C2311yescharacteristic
216#107C22×C2311yescharacteristic
217#107C22×C2311yescharacteristic
218#106C22×C2311yescharacteristic
219#106C22×C2311yescharacteristic
220#106C22×C2311yescharacteristic
221#106C22×C2311yescharacteristic
222C4⋊C8C22×C2311yescharacteristic
223#102C22×C2311yescharacteristic
224#102C22×C2311yescharacteristic
225#102C22×C2311yescharacteristic
226#102C22×C2311yescharacteristic
227#102C22×C2311yescharacteristic
228#102C22×C2311yescharacteristic
229C22⋊C8C22×C2311yescharacteristic
230C22⋊C8C22×C2311yescharacteristic
231#97C22×C2311yescharacteristic
232C23#334711yescharacteristic, direct product factor, Sylow
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4D4×C2311yescharacteristic
239C2×D4C23×C2311yescharacteristic
240C2×D4C23×C2311yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4C23×C2311yescharacteristic
245C22×C4C23×C2311yescharacteristic
246D8—22no—
247M16—22no—
248C2×C8—22no—
249C2×C8C23×C2311yescharacteristic
250C2×C8C23×C2311yescharacteristic
251C2×C8C23×C2311yescharacteristic
252C2×C8C23×C2311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4D4×C2311yescharacteristic
258C4⋊C4C23×C2311yescharacteristic
259C4⋊C4C23×C2311yescharacteristic
260C4⋊C4C23×C2311yescharacteristic
261C4⋊C4C23×C2311yescharacteristic
262C4⋊C4C23×C2311yescharacteristic
263C4⋊C4C23×C2311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4D4×C2311yescharacteristic
274C22⋊C4D4×C2311yescharacteristic
275C42—21no—
276C42—21no—
277C42—21no—
278C42C23×C2311yescharacteristic
279C23—21no—
280C23—21no—
281C23C2×D4×C2311yescharacteristic
282C23C2×D4×C2311yescharacteristic
283Q8—81no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—41no—
289D4—42no—
290D4—81no—
291D4—81no—
292D4—81no—
293C2×C4—21no—
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—42no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4C2×D4×C2311yescharacteristic
307C2×C4C2×D4×C2311yescharacteristic
308C2×C4C2×D4×C2311yescharacteristic
309C2×C4C2×D4×C2311yescharacteristic
310C2×C4C24×C2311yescharacteristic
311C8—21no—
312C8—21no—
313C8—21no—
314C8#41×C2312yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22C22×D4×C2311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4C2×#41×C2311yescharacteristic
339C4C23×2−1+411yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—81no—
344C2C23×#54711yescharacteristic
345C2C23×#57511yescharacteristic
346C2C23×#57611yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#555C2semidirect
2. C23×#553C2semidirect
3. C23×#501C2semidirect
4. C23×#482C2semidirect
5. C23×#482C2semidirect
6. C23×#482C2semidirect
7. C23×#480C2semidirect
8. C23×#478C2semidirect
9. C23×#474C2semidirect
10. C23×#469C2semidirect
11. C23×#466C2semidirect
12. C23×#442C2semidirect
13. C23×#435C2semidirect
14. C23×#428C2semidirect
15. C23×#427C2semidirect
141. #3347C23direct (N × H)
167. #555C2×C23semidirect
168. #553C2×C23semidirect
169. #501C2×C23semidirect
170. #482C2×C23semidirect
171. #482C2×C23semidirect
172. #482C2×C23semidirect
173. #480C2×C23semidirect
174. #478C2×C23semidirect
175. #474C2×C23semidirect
176. #469C2×C23semidirect
177. #466C2×C23semidirect
178. #442C2×C23semidirect
179. #435C2×C23semidirect
180. #428C2×C23semidirect
181. #427C2×C23semidirect
232. C23#3347direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5552
C23×#5532
C23×#5012
C23×#4822
C23×#4802
C23×#4782
C23×#4742
C23×#4692
C23×#4662
C23×#4422
C23×#4352
C23×#4282
C23×#4272
#334723