Finite Group Databasebeta

← C23×#3201 #15670820

Subgroups — C23×#3201 #15670820

Order 2944.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#537C211yesmaximal, characteristic
2C23×#536C211yesmaximal, characteristic, complemented
3C23×#485C211yesmaximal, characteristic, complemented
4C23×#484C211yesmaximal, characteristic, complemented
5C23×#483C211yesmaximal, characteristic, complemented
6C23×#482C211yesmaximal, characteristic, complemented
7C23×#470C211yesmaximal, characteristic, complemented
8C23×#468C211yesmaximal, characteristic, complemented
9C23×#467C211yesmaximal, characteristic, complemented
10C23×#465C211yesmaximal, characteristic, complemented
11C23×#452C211yesmaximal, characteristic, complemented
12C23×#452C211yesmaximal, characteristic, complemented
13C23×#451C211yesmaximal, characteristic, complemented
14C23×#448C211yesmaximal, characteristic, complemented
15C23×#408C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic
18C22×Q8×C23C2211yescharacteristic
19C2×Q16×C23C2211yescharacteristic
20C2×Q16×C23C2211yescharacteristic
21C2×SD16×C23C2211yescharacteristic
22C2×SD16×C23C2211yescharacteristic
23C2×M16×C23C2211yescharacteristic
24C22×C8×C23C2211yescharacteristic
25C23×C4⋊Q8—21no—
26C23×#124—21no—
27C23×#123—21no—
28C23×#123—21no—
29C23×C22⋊Q8—21no—
30C23×C22⋊Q8C2211yescharacteristic
31C23×C22⋊Q8C2211yescharacteristic
32C23×C22⋊Q8C2211yescharacteristic
33C23×C22⋊Q8C2211yescharacteristic
34C23×C22⋊Q8C2211yescharacteristic
35C23×C22⋊Q8C2211yescharacteristic
36C23×#121—21no—
37C23×#121—21no—
38C23×#121—21no—
39C23×#121C2211yescharacteristic
40C23×#121C2211yescharacteristic
41C23×#117C2211yescharacteristic
42C2×C4⋊C4×C23C2211yescharacteristic
43C23×#107C2211yescharacteristic
44C23×#107C2211yescharacteristic
45C23×#106C2211yescharacteristic
46C23×#106C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×#103C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×C22⋊C8C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C23×C22⋊C8C2211yescharacteristic
59#41×C23—41no—
60#41×C23—41no—
61#41×C23—41no—
62#41×C23—41no—
63C2×Q8×C23—21no—
64C2×Q8×C23—41no—
65C2×Q8×C23—41no—
66C2×Q8×C23D411yescharacteristic
67C2×Q8×C23C2311yescharacteristic
68C2×Q8×C23C2311yescharacteristic
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—21no—
73C2×D4×C23D411yescharacteristic
74C2×D4×C23D411yescharacteristic
75C2×D4×C23D411yescharacteristic
76C2×D4×C23C2311yescharacteristic
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23C2311yescharacteristic
81C22×C4×C23C2311yescharacteristic
82C22×C4×C23C2311yescharacteristic
83Q16×C23—41no—
84Q16×C23—41no—
85SD16×C23—41no—
86SD16×C23—41no—
87M16×C23—41no—
88C2×C8×C23—41no—
89C2×C8×C23C2311yescharacteristic
90C2×C8×C23C2311yescharacteristic
91C2×C8×C23C2311yescharacteristic
92C2×C8×C23C2311yescharacteristic
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23C2311yescharacteristic
101C4⋊C4×C23C2311yescharacteristic
102C4⋊C4×C23C2311yescharacteristic
103C4⋊C4×C23C2311yescharacteristic
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23—21no—
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—21no—
110C22⋊C4×C23—21no—
111C22⋊C4×C23—21no—
112C22⋊C4×C23—21no—
113C42×C23—21no—
114C23×C23—21no—
115C23×C23C2×D411yescharacteristic
116C23×C23C2×D411yescharacteristic
117C23×C23C2×D411yescharacteristic
118Q8×C23—21no—
119Q8×C23—21no—
120Q8×C23—21no—
121Q8×C23—21no—
122Q8×C23—21no—
123Q8×C23—21no—
124Q8×C23—41no—
125Q8×C23—42no—
126D4×C23—21no—
127D4×C23—21no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—42no—
132D4×C23—81no—
133D4×C23—81no—
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—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—81no—
150C2×C4×C23C2×D411yescharacteristic
151C2×C4×C23C2×D411yescharacteristic
152C2×C4×C23C2×D411yescharacteristic
153C2×C4×C23C2411yescharacteristic
154C8×C23—21no—
155C8×C23—21no—
156C8×C23—21no—
157C8×C23—21no—
158#3201C2311yesmaximal, characteristic, direct product factor, Sylow
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—41no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23C22×D411yescharacteristic
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—41no—
176C4×C23—41no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23H(5,𝔽2)11yescharacteristic
183C4×C23H(5,𝔽2)11yescharacteristic
184#537C2×C2311yescharacteristic
185#536C2×C2311yescharacteristic, complemented
186#485C2×C2311yescharacteristic, complemented
187#484C2×C2311yescharacteristic, complemented
188#483C2×C2311yescharacteristic, complemented
189#482C2×C2311yescharacteristic, complemented
190#470C2×C2311yescharacteristic, complemented
191#468C2×C2311yescharacteristic, complemented
192#467C2×C2311yescharacteristic, complemented
193#465C2×C2311yescharacteristic, complemented
194#452C2×C2311yescharacteristic, complemented
195#452C2×C2311yescharacteristic, complemented
196#451C2×C2311yescharacteristic, complemented
197#448C2×C2311yescharacteristic, complemented
198#408C2×C2311yescharacteristic, complemented
199C2×C23—41no—
200C2×C23—41no—
201C2×C23—41no—
202C2×C23—81no—
203C2×C23#53411yescharacteristic
204C2×C23#57711yescharacteristic
205C2×C23#57811yescharacteristic
206C2×#41C22×C2311yescharacteristic
207C2×#41C22×C2311yescharacteristic
208C22×Q8C22×C2311yescharacteristic
209C2×Q16C22×C2311yescharacteristic
210C2×Q16C22×C2311yescharacteristic
211C2×SD16C22×C2311yescharacteristic
212C2×SD16C22×C2311yescharacteristic
213C2×M16C22×C2311yescharacteristic
214C22×C8C22×C2311yescharacteristic
215C4⋊Q8—21no—
216#124—21no—
217#123—21no—
218#123—21no—
219C22⋊Q8—21no—
220C22⋊Q8C22×C2311yescharacteristic
221C22⋊Q8C22×C2311yescharacteristic
222C22⋊Q8C22×C2311yescharacteristic
223C22⋊Q8C22×C2311yescharacteristic
224C22⋊Q8C22×C2311yescharacteristic
225C22⋊Q8C22×C2311yescharacteristic
226#121—21no—
227#121—21no—
228#121—21no—
229#121C22×C2311yescharacteristic
230#121C22×C2311yescharacteristic
231#117C22×C2311yescharacteristic
232C2×C4⋊C4C22×C2311yescharacteristic
233#107C22×C2311yescharacteristic
234#107C22×C2311yescharacteristic
235#106C22×C2311yescharacteristic
236#106C22×C2311yescharacteristic
237#103C22×C2311yescharacteristic
238#103C22×C2311yescharacteristic
239#103C22×C2311yescharacteristic
240#103C22×C2311yescharacteristic
241#103C22×C2311yescharacteristic
242#103C22×C2311yescharacteristic
243#102C22×C2311yescharacteristic
244#102C22×C2311yescharacteristic
245C22⋊C8C22×C2311yescharacteristic
246C22⋊C8C22×C2311yescharacteristic
247C22⋊C8C22×C2311yescharacteristic
248C22⋊C8C22×C2311yescharacteristic
249C23#320111yescharacteristic, direct product factor, Sylow
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8D4×C2311yescharacteristic
258C2×Q8C23×C2311yescharacteristic
259C2×Q8C23×C2311yescharacteristic
260C2×Q8C23×C2311yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4D4×C2311yescharacteristic
265C2×D4D4×C2311yescharacteristic
266C2×D4D4×C2311yescharacteristic
267C2×D4C23×C2311yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C23×C2311yescharacteristic
272C22×C4C23×C2311yescharacteristic
273C22×C4C23×C2311yescharacteristic
274Q16—41no—
275Q16—41no—
276SD16—41no—
277SD16—41no—
278M16—41no—
279C2×C8—41no—
280C2×C8C23×C2311yescharacteristic
281C2×C8C23×C2311yescharacteristic
282C2×C8C23×C2311yescharacteristic
283C2×C8C23×C2311yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4C23×C2311yescharacteristic
292C4⋊C4C23×C2311yescharacteristic
293C4⋊C4C23×C2311yescharacteristic
294C4⋊C4C23×C2311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C23—21no—
306C23C2×D4×C2311yescharacteristic
307C23C2×D4×C2311yescharacteristic
308C23C2×D4×C2311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—41no—
316Q8—42no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4C2×D4×C2311yescharacteristic
342C2×C4C2×D4×C2311yescharacteristic
343C2×C4C2×D4×C2311yescharacteristic
344C2×C4C24×C2311yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×D4×C2311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4C23×H(5,𝔽2)11yescharacteristic
373C4C23×H(5,𝔽2)11yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2C23×#53411yescharacteristic
379C2C23×#57711yescharacteristic
380C2C23×#57811yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#536C2semidirect
3. C23×#485C2semidirect
4. C23×#484C2semidirect
5. C23×#483C2semidirect
6. C23×#482C2semidirect
7. C23×#470C2semidirect
8. C23×#468C2semidirect
9. C23×#467C2semidirect
10. C23×#465C2semidirect
11. C23×#452C2semidirect
12. C23×#452C2semidirect
13. C23×#451C2semidirect
14. C23×#448C2semidirect
15. C23×#408C2semidirect
158. #3201C23direct (N × H)
185. #536C2×C23semidirect
186. #485C2×C23semidirect
187. #484C2×C23semidirect
188. #483C2×C23semidirect
189. #482C2×C23semidirect
190. #470C2×C23semidirect
191. #468C2×C23semidirect
192. #467C2×C23semidirect
193. #465C2×C23semidirect
194. #452C2×C23semidirect
195. #452C2×C23semidirect
196. #451C2×C23semidirect
197. #448C2×C23semidirect
198. #408C2×C23semidirect
249. C23#3201direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5372
C23×#5362
C23×#4852
C23×#4842
C23×#4832
C23×#4822
C23×#4702
C23×#4682
C23×#4672
C23×#4652
C23×#4522
C23×#4512
C23×#4482
C23×#4082
#320123