Finite Group Databasebeta

← C23×#3321 #15670940

Subgroups — C23×#3321 #15670940

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#555C211yesmaximal, characteristic, complemented
2C23×#548C211yesmaximal, characteristic, complemented
3C23×#489C211yesmaximal, characteristic, complemented
4C23×#485C211yesmaximal, characteristic, complemented
5C23×#482C211yesmaximal, characteristic, complemented
6C23×#475C211yesmaximal, characteristic, complemented
7C23×#469C211yesmaximal, characteristic, complemented
8C23×#465C211yesmaximal, characteristic, complemented
9C23×#463C211yesmaximal, characteristic, complemented
10C23×#449C211yesmaximal, characteristic, complemented
11C23×#448C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#419C211yesmaximal, characteristic, complemented
15C23×#416C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×D8×C23C2211yescharacteristic
21C2×M16×C23C2211yescharacteristic
22C22×C8×C23C2211yescharacteristic
23C23×C4⋊Q8C2211yescharacteristic
24C23×#124—21no—
25C23×#124—21no—
26C23×#124—21no—
27C23×#124C2211yescharacteristic
28C23×C22⋊Q8—21no—
29C23×C22⋊Q8C2211yescharacteristic
30C23×#121—21no—
31C23×#121—21no—
32C23×#121C2211yescharacteristic
33C23×#121C2211yescharacteristic
34C23×#121C2211yescharacteristic
35C4×Q8×C23C2211yescharacteristic
36C4×D4×C23—21no—
37C4×D4×C23C2211yescharacteristic
38C4×D4×C23C2211yescharacteristic
39C23×#117—21no—
40C23×#117C2211yescharacteristic
41C23×#117C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#106C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×#103C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C23×#97C2211yescharacteristic
59#41×C23—22no—
60#41×C23—41no—
61C2×Q8×C23—21no—
62C2×Q8×C23—21no—
63C2×Q8×C23—22no—
64C2×Q8×C23—41no—
65C2×Q8×C23C2311yescharacteristic
66C2×Q8×C23C2311yescharacteristic
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23D411yescharacteristic
72C2×D4×C23C2311yescharacteristic
73C2×D4×C23C2311yescharacteristic
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23C2311yescharacteristic
78C22×C4×C23C2311yescharacteristic
79SD16×C23—22no—
80SD16×C23—41no—
81D8×C23—41no—
82M16×C23—41no—
83C2×C8×C23—41no—
84C2×C8×C23C2311yescharacteristic
85C2×C8×C23C2311yescharacteristic
86C2×C8×C23C2311yescharacteristic
87C2×C8×C23C2311yescharacteristic
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—21no—
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×C23—21no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23—21no—
103C22⋊C4×C23—21no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23D411yescharacteristic
106C22⋊C4×C23D411yescharacteristic
107C42×C23—21no—
108C42×C23—21no—
109C42×C23—21no—
110C42×C23C2311yescharacteristic
111C23×C23—21no—
112C23×C23—21no—
113C23×C23C2×D411yescharacteristic
114C23×C23C2×D411yescharacteristic
115Q8×C23—21no—
116Q8×C23—21no—
117Q8×C23—21no—
118Q8×C23—42no—
119Q8×C23—81no—
120Q8×C23#4112yes—
121D4×C23—21no—
122D4×C23—21no—
123D4×C23—21no—
124D4×C23—21no—
125D4×C23—41no—
126D4×C23—42no—
127D4×C23—42no—
128D4×C23—81no—
129D4×C23—81no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—42no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23C2×D411yescharacteristic
144C2×C4×C23C2×D411yescharacteristic
145C2×C4×C23C2×D411yescharacteristic
146C2×C4×C23C2×D411yescharacteristic
147C2×C4×C23C2411yescharacteristic
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—21no—
151C8×C23—21no—
152#3321C2311yesmaximal, characteristic, direct product factor, Sylow
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—41no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23—41no—
165C22×C23C22×D411yescharacteristic
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—22no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23—41no—
176C4×C23C2×#4111yescharacteristic
177C4×C23H(5,𝔽2)11yescharacteristic
178#555C2×C2311yescharacteristic, complemented
179#548C2×C2311yescharacteristic, complemented
180#489C2×C2311yescharacteristic, complemented
181#485C2×C2311yescharacteristic, complemented
182#482C2×C2311yescharacteristic, complemented
183#475C2×C2311yescharacteristic, complemented
184#469C2×C2311yescharacteristic, complemented
185#465C2×C2311yescharacteristic, complemented
186#463C2×C2311yescharacteristic, complemented
187#449C2×C2311yescharacteristic, complemented
188#448C2×C2311yescharacteristic, complemented
189#440C2×C2311yescharacteristic, complemented
190#435C2×C2311yescharacteristic, complemented
191#419C2×C2311yescharacteristic, complemented
192#416C2×C2311yescharacteristic, complemented
193C2×C23—41no—
194C2×C23—41no—
195C2×C23—81no—
196C2×C23—81no—
197C2×C23#54611yescharacteristic
198C2×C23#57511yescharacteristic
199C2×C23#57711yescharacteristic
200C2×#41C22×C2311yescharacteristic
201C22×Q8C22×C2311yescharacteristic
202C2×SD16C22×C2311yescharacteristic
203C2×SD16C22×C2311yescharacteristic
204C2×D8C22×C2311yescharacteristic
205C2×M16C22×C2311yescharacteristic
206C22×C8C22×C2311yescharacteristic
207C4⋊Q8C22×C2311yescharacteristic
208#124—21no—
209#124—21no—
210#124—21no—
211#124C22×C2311yescharacteristic
212C22⋊Q8—21no—
213C22⋊Q8C22×C2311yescharacteristic
214#121—21no—
215#121—21no—
216#121C22×C2311yescharacteristic
217#121C22×C2311yescharacteristic
218#121C22×C2311yescharacteristic
219C4×Q8C22×C2311yescharacteristic
220C4×D4—21no—
221C4×D4C22×C2311yescharacteristic
222C4×D4C22×C2311yescharacteristic
223#117—21no—
224#117C22×C2311yescharacteristic
225#117C22×C2311yescharacteristic
226#107C22×C2311yescharacteristic
227#106C22×C2311yescharacteristic
228#106C22×C2311yescharacteristic
229C4⋊C8C22×C2311yescharacteristic
230#103C22×C2311yescharacteristic
231#103C22×C2311yescharacteristic
232#103C22×C2311yescharacteristic
233#103C22×C2311yescharacteristic
234#103C22×C2311yescharacteristic
235#103C22×C2311yescharacteristic
236#102C22×C2311yescharacteristic
237#102C22×C2311yescharacteristic
238#102C22×C2311yescharacteristic
239#102C22×C2311yescharacteristic
240C22⋊C8C22×C2311yescharacteristic
241C22⋊C8C22×C2311yescharacteristic
242#97C22×C2311yescharacteristic
243C23#332111yescharacteristic, direct product factor, Sylow
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8C23×C2311yescharacteristic
251C2×Q8C23×C2311yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4D4×C2311yescharacteristic
257C2×D4C23×C2311yescharacteristic
258C2×D4C23×C2311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4C23×C2311yescharacteristic
263C22×C4C23×C2311yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—41no—
267M16—41no—
268C2×C8—41no—
269C2×C8C23×C2311yescharacteristic
270C2×C8C23×C2311yescharacteristic
271C2×C8C23×C2311yescharacteristic
272C2×C8C23×C2311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4D4×C2311yescharacteristic
277C4⋊C4C23×C2311yescharacteristic
278C4⋊C4C23×C2311yescharacteristic
279C4⋊C4C23×C2311yescharacteristic
280C4⋊C4C23×C2311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4D4×C2311yescharacteristic
291C22⋊C4D4×C2311yescharacteristic
292C42—21no—
293C42—21no—
294C42—21no—
295C42C23×C2311yescharacteristic
296C23—21no—
297C23—21no—
298C23C2×D4×C2311yescharacteristic
299C23C2×D4×C2311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8—81no—
305Q8#41×C2312yes—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—42no—
312D4—42no—
313D4—81no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4C2×D4×C2311yescharacteristic
329C2×C4C2×D4×C2311yescharacteristic
330C2×C4C2×D4×C2311yescharacteristic
331C2×C4C2×D4×C2311yescharacteristic
332C2×C4C24×C2311yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C22×D4×C2311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4C2×#41×C2311yescharacteristic
361C4C23×H(5,𝔽2)11yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2C23×#54611yescharacteristic
367C2C23×#57511yescharacteristic
368C2C23×#57711yescharacteristic
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×#548C2semidirect
3. C23×#489C2semidirect
4. C23×#485C2semidirect
5. C23×#482C2semidirect
6. C23×#475C2semidirect
7. C23×#469C2semidirect
8. C23×#465C2semidirect
9. C23×#463C2semidirect
10. C23×#449C2semidirect
11. C23×#448C2semidirect
12. C23×#440C2semidirect
13. C23×#435C2semidirect
14. C23×#419C2semidirect
15. C23×#416C2semidirect
152. #3321C23direct (N × H)
178. #555C2×C23semidirect
179. #548C2×C23semidirect
180. #489C2×C23semidirect
181. #485C2×C23semidirect
182. #482C2×C23semidirect
183. #475C2×C23semidirect
184. #469C2×C23semidirect
185. #465C2×C23semidirect
186. #463C2×C23semidirect
187. #449C2×C23semidirect
188. #448C2×C23semidirect
189. #440C2×C23semidirect
190. #435C2×C23semidirect
191. #419C2×C23semidirect
192. #416C2×C23semidirect
243. C23#3321direct (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×#5482
C23×#4892
C23×#4852
C23×#4822
C23×#4752
C23×#4692
C23×#4652
C23×#4632
C23×#4492
C23×#4482
C23×#4402
C23×#4352
C23×#4192
C23×#4162
#332123