Finite Group Databasebeta

← C23×#3320 #15670939

Subgroups — C23×#3320 #15670939

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#553C211yesmaximal, characteristic
2D4×Q8×C23C211yesmaximal, characteristic, complemented
3C23×#491C211yesmaximal, characteristic, complemented
4C23×#485C211yesmaximal, characteristic, complemented
5C23×#484C211yesmaximal, characteristic, complemented
6C23×#479C211yesmaximal, characteristic, complemented
7C23×#468C211yesmaximal, characteristic, complemented
8C23×#465C211yesmaximal, characteristic, complemented
9C23×#461C211yesmaximal, characteristic, complemented
10C23×#448C211yesmaximal, characteristic, complemented
11C23×#448C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#419C211yesmaximal, characteristic, complemented
15C2×C23×#103C211yesmaximal, characteristic, complemented
16C22×Q8×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×SD16×C23C2211yescharacteristic
21C2×M16×C23C2211yescharacteristic
22C22×C8×C23C2211yescharacteristic
23C23×C4⋊Q8—21no—
24C23×C4⋊Q8C2211yescharacteristic
25C23×#126—21no—
26C23×#125C2211yescharacteristic
27C23×#123—21no—
28C23×C22⋊Q8—21no—
29C23×C22⋊Q8—21no—
30C23×C22⋊Q8C2211yescharacteristic
31C23×C22⋊Q8C2211yescharacteristic
32C23×#121—21no—
33C23×#121C2211yescharacteristic
34C23×#121C2211yescharacteristic
35C4×Q8×C23C2211yescharacteristic
36C4×D4×C23—21no—
37C4×D4×C23—21no—
38C4×D4×C23C2211yescharacteristic
39C4×D4×C23C2211yescharacteristic
40C23×#117C2211yescharacteristic
41C2×C4⋊C4×C23C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#107C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×#103C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×#103C2211yescharacteristic
53C23×#103C2211yescharacteristic
54C23×#103C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C23×#97C2211yescharacteristic
59C2×Q8×C23—21no—
60C2×Q8×C23—21no—
61C2×Q8×C23—22no—
62C2×Q8×C23—22no—
63C2×Q8×C23—41no—
64C2×Q8×C23—41no—
65C2×Q8×C23C2311yescharacteristic
66C2×Q8×C23C2311yescharacteristic
67C2×Q8×C23C2311yescharacteristic
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23D411yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23C2311yescharacteristic
77C22×C4×C23C2311yescharacteristic
78SD16×C23—22no—
79SD16×C23—41no—
80SD16×C23—41no—
81M16×C23—41no—
82C2×C8×C23—41no—
83C2×C8×C23C2311yescharacteristic
84C2×C8×C23C2311yescharacteristic
85C2×C8×C23C2311yescharacteristic
86C2×C8×C23C2311yescharacteristic
87C4⋊C4×C23—21no—
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—21no—
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23D411yescharacteristic
96C4⋊C4×C23C2311yescharacteristic
97C4⋊C4×C23C2311yescharacteristic
98C4⋊C4×C23C2311yescharacteristic
99C4⋊C4×C23C2311yescharacteristic
100C22⋊C4×C23—21no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23—21no—
103C22⋊C4×C23—21no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23D411yescharacteristic
107C22⋊C4×C23D411yescharacteristic
108C42×C23—21no—
109C42×C23—21no—
110C42×C23C2311yescharacteristic
111C23×C23—21no—
112C23×C23C2×D411yescharacteristic
113C23×C23C2×D411yescharacteristic
114Q8×C23—21no—
115Q8×C23—21no—
116Q8×C23—21no—
117Q8×C23—21no—
118Q8×C23—21no—
119Q8×C23—42no—
120Q8×C23—42no—
121Q8×C23#4112yes—
122D4×C23—22no—
123D4×C23—21no—
124D4×C23—21no—
125D4×C23—42no—
126D4×C23—81no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
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—42no—
139C2×C4×C23—42no—
140C2×C4×C23—81no—
141C2×C4×C23—81no—
142C2×C4×C23C2×D411yescharacteristic
143C2×C4×C23C2×D411yescharacteristic
144C2×C4×C23C2×D411yescharacteristic
145C2×C4×C23C2×D411yescharacteristic
146C2×C4×C23C2411yescharacteristic
147C8×C23—21no—
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—21no—
151#3320C2311yesmaximal, characteristic, direct product factor, Sylow
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23#13712yes—
161C22×C23C22×D411yescharacteristic
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—22no—
165C4×C23—21no—
166C4×C23—21no—
167C4×C23—41no—
168C4×C23—41no—
169C4×C23—41no—
170C4×C23—41no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23C2×#4111yescharacteristic
174C4×C23H(5,𝔽2)11yescharacteristic
175#553C2×C2311yescharacteristic
176D4×Q8C2×C2311yescharacteristic, complemented
177#491C2×C2311yescharacteristic, complemented
178#485C2×C2311yescharacteristic, complemented
179#484C2×C2311yescharacteristic, complemented
180#479C2×C2311yescharacteristic, complemented
181#468C2×C2311yescharacteristic, complemented
182#465C2×C2311yescharacteristic, complemented
183#461C2×C2311yescharacteristic, complemented
184#448C2×C2311yescharacteristic, complemented
185#448C2×C2311yescharacteristic, complemented
186#440C2×C2311yescharacteristic, complemented
187#435C2×C2311yescharacteristic, complemented
188#419C2×C2311yescharacteristic, complemented
189C2×#103C2×C2311yescharacteristic, complemented
190C2×C23—22no—
191C2×C23—41no—
192C2×C23—81no—
193C2×C23#54611yescharacteristic
194C2×C23C2×#13711yescharacteristic
195C2×C23#57711yescharacteristic
196C22×Q8C22×C2311yescharacteristic
197C22×Q8C22×C2311yescharacteristic
198C2×SD16C22×C2311yescharacteristic
199C2×SD16C22×C2311yescharacteristic
200C2×SD16C22×C2311yescharacteristic
201C2×M16C22×C2311yescharacteristic
202C22×C8C22×C2311yescharacteristic
203C4⋊Q8—21no—
204C4⋊Q8C22×C2311yescharacteristic
205#126—21no—
206#125C22×C2311yescharacteristic
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8C22×C2311yescharacteristic
211C22⋊Q8C22×C2311yescharacteristic
212#121—21no—
213#121C22×C2311yescharacteristic
214#121C22×C2311yescharacteristic
215C4×Q8C22×C2311yescharacteristic
216C4×D4—21no—
217C4×D4—21no—
218C4×D4C22×C2311yescharacteristic
219C4×D4C22×C2311yescharacteristic
220#117C22×C2311yescharacteristic
221C2×C4⋊C4C22×C2311yescharacteristic
222#107C22×C2311yescharacteristic
223#107C22×C2311yescharacteristic
224#106C22×C2311yescharacteristic
225C4⋊C8C22×C2311yescharacteristic
226#103C22×C2311yescharacteristic
227#103C22×C2311yescharacteristic
228#103C22×C2311yescharacteristic
229#103C22×C2311yescharacteristic
230#103C22×C2311yescharacteristic
231#103C22×C2311yescharacteristic
232#103C22×C2311yescharacteristic
233#103C22×C2311yescharacteristic
234#103C22×C2311yescharacteristic
235#102C22×C2311yescharacteristic
236C22⋊C8C22×C2311yescharacteristic
237C22⋊C8C22×C2311yescharacteristic
238#97C22×C2311yescharacteristic
239C23#332011yescharacteristic, direct product factor, Sylow
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C23×C2311yescharacteristic
247C2×Q8C23×C2311yescharacteristic
248C2×Q8C23×C2311yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4D4×C2311yescharacteristic
252C2×D4C23×C2311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4C23×C2311yescharacteristic
258C22×C4C23×C2311yescharacteristic
259SD16—22no—
260SD16—41no—
261SD16—41no—
262M16—41no—
263C2×C8—41no—
264C2×C8C23×C2311yescharacteristic
265C2×C8C23×C2311yescharacteristic
266C2×C8C23×C2311yescharacteristic
267C2×C8C23×C2311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
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⋊C4D4×C2311yescharacteristic
288C22⋊C4D4×C2311yescharacteristic
289C42—21no—
290C42—21no—
291C42C23×C2311yescharacteristic
292C23—21no—
293C23C2×D4×C2311yescharacteristic
294C23C2×D4×C2311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#41×C2312yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×D4×C2311yescharacteristic
324C2×C4C2×D4×C2311yescharacteristic
325C2×C4C2×D4×C2311yescharacteristic
326C2×C4C2×D4×C2311yescharacteristic
327C2×C4C24×C2311yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22C23×#13712yes—
341C22C22×D4×C2311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4C2×#41×C2311yescharacteristic
354C4C23×H(5,𝔽2)11yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2C23×#54611yescharacteristic
359C2C2×C23×#13711yescharacteristic
360C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#553C2
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
2. D4×Q8×C23C2semidirect
3. C23×#491C2semidirect
4. C23×#485C2semidirect
5. C23×#484C2semidirect
6. C23×#479C2semidirect
7. C23×#468C2semidirect
8. C23×#465C2semidirect
9. C23×#461C2semidirect
10. C23×#448C2semidirect
11. C23×#448C2semidirect
12. C23×#440C2semidirect
13. C23×#435C2semidirect
14. C23×#419C2semidirect
15. C2×C23×#103C2semidirect
151. #3320C23direct (N × H)
176. D4×Q8C2×C23semidirect
177. #491C2×C23semidirect
178. #485C2×C23semidirect
179. #484C2×C23semidirect
180. #479C2×C23semidirect
181. #468C2×C23semidirect
182. #465C2×C23semidirect
183. #461C2×C23semidirect
184. #448C2×C23semidirect
185. #448C2×C23semidirect
186. #440C2×C23semidirect
187. #435C2×C23semidirect
188. #419C2×C23semidirect
189. C2×#103C2×C23semidirect
239. C23#3320direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5532
D4×Q8×C232
C23×#4912
C23×#4852
C23×#4842
C23×#4792
C23×#4682
C23×#4652
C23×#4612
C23×#4482
C23×#4402
C23×#4352
C23×#4192
C2×C23×#1032
#332023