Finite Group Databasebeta

← C23×#3178 #15670797

Subgroups — C23×#3178 #15670797

Order 2944.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#542C211yesmaximal, characteristic
2C23×#541C211yesmaximal, characteristic, complemented
3C23×#501C211yesmaximal, characteristic, complemented
4C23×#497C211yesmaximal, characteristic, complemented
5C23×#488C211yesmaximal, characteristic, complemented
6C23×#487C211yesmaximal, characteristic, complemented
7C23×#470C211yesmaximal, characteristic, complemented
8C23×#468C211yesmaximal, characteristic
9C23×#467C211yesmaximal, characteristic, complemented
10C23×#465C211yesmaximal, characteristic
11C23×#441C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C4×Q16×C23C211yesmaximal, characteristic, complemented
14C4×SD16×C23C211yesmaximal, characteristic, complemented
15C23×#405C211yesmaximal, characteristic, complemented
16C2×Q16×C23C2211yescharacteristic
17C2×Q16×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×M16×C23C2211yescharacteristic
21C22×C8×C23C2211yescharacteristic
22C23×C4⋊Q8C2211yescharacteristic
23C23×C4⋊Q8C2211yescharacteristic
24C23×#126—21no—
25C23×#126—21no—
26C23×#126—21no—
27C23×#125—21no—
28C23×#125—21no—
29C23×#125C2211yescharacteristic
30C23×#124—21no—
31C23×#124C2211yescharacteristic
32C23×#123—21no—
33C23×C22⋊Q8—21no—
34C23×C22⋊Q8C2211yescharacteristic
35C23×C22⋊Q8C2211yescharacteristic
36C23×C22⋊Q8C2211yescharacteristic
37C23×#121C2211yescharacteristic
38C4×Q8×C23C2211yescharacteristic
39C4×Q8×C23C2211yescharacteristic
40C4×Q8×C23C2211yescharacteristic
41C4×D4×C23C2211yescharacteristic
42C23×#117C2211yescharacteristic
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×#97C2211yescharacteristic
56C23×#97C2211yescharacteristic
57C4×C8×C23C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59C2×Q8×C23—21no—
60C2×Q8×C23C2311yescharacteristic
61C2×Q8×C23C2311yescharacteristic
62C2×Q8×C23C2311yescharacteristic
63C2×D4×C23—21no—
64C2×D4×C23C2311yescharacteristic
65C22×C4×C23—21no—
66C22×C4×C23C2311yescharacteristic
67Q16×C23—22no—
68Q16×C23—22no—
69SD16×C23—22no—
70SD16×C23—22no—
71M16×C23—22no—
72C2×C8×C23—22no—
73C2×C8×C23C2311yescharacteristic
74C2×C8×C23C2311yescharacteristic
75C2×C8×C23C2311yescharacteristic
76C2×C8×C23C2311yescharacteristic
77C4⋊C4×C23—21no—
78C4⋊C4×C23—21no—
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×C23—21no—
86C4⋊C4×C23—21no—
87C4⋊C4×C23—21no—
88C4⋊C4×C23D411yescharacteristic
89C4⋊C4×C23D411yescharacteristic
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C4⋊C4×C23C2311yescharacteristic
93C4⋊C4×C23C2311yescharacteristic
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—
101C22⋊C4×C23D411yescharacteristic
102C22⋊C4×C23D411yescharacteristic
103C42×C23—21no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23C2311yescharacteristic
107C42×C23C2311yescharacteristic
108C23×C23—21no—
109C23×C23C2×D411yescharacteristic
110Q8×C23—21no—
111Q8×C23—21no—
112Q8×C23—21no—
113Q8×C23—21no—
114Q8×C23—21no—
115Q8×C23—21no—
116Q8×C23—81no—
117D4×C23—21no—
118D4×C23—21no—
119D4×C23—81no—
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—81no—
130C2×C4×C23C2×D411yescharacteristic
131C2×C4×C23C2×D411yescharacteristic
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2×D411yescharacteristic
134C2×C4×C23C2×D411yescharacteristic
135C2×C4×C23C2411yescharacteristic
136C8×C23—21no—
137C8×C23—21no—
138C8×C23#4112yes—
139C8×C23#4112yes—
140#3178C2311yesmaximal, characteristic, direct product factor, Sylow
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23C22×D411yescharacteristic
148C4×C23—21no—
149C4×C23—21no—
150C4×C23—21no—
151C4×C23—21no—
152C4×C23—21no—
153C4×C23—41no—
154C4×C23—41no—
155C4×C23—41no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23C2×#4111yescharacteristic
161C4×C23C2×#4111yescharacteristic
162#542C2×C2311yescharacteristic
163#541C2×C2311yescharacteristic, complemented
164#501C2×C2311yescharacteristic, complemented
165#497C2×C2311yescharacteristic, complemented
166#488C2×C2311yescharacteristic, complemented
167#487C2×C2311yescharacteristic, complemented
168#470C2×C2311yescharacteristic, complemented
169#468C2×C2311yescharacteristic
170#467C2×C2311yescharacteristic, complemented
171#465C2×C2311yescharacteristic
172#441C2×C2311yescharacteristic, complemented
173#440C2×C2311yescharacteristic, complemented
174C4×Q16C2×C2311yescharacteristic, complemented
175C4×SD16C2×C2311yescharacteristic, complemented
176#405C2×C2311yescharacteristic, complemented
177C2×C23—41no—
178C2×C23—81no—
179C2×C23#53211yescharacteristic
180C2×C23#57711yescharacteristic
181C2×C23#57811yescharacteristic
182C2×Q16C22×C2311yescharacteristic
183C2×Q16C22×C2311yescharacteristic
184C2×SD16C22×C2311yescharacteristic
185C2×SD16C22×C2311yescharacteristic
186C2×M16C22×C2311yescharacteristic
187C22×C8C22×C2311yescharacteristic
188C4⋊Q8C22×C2311yescharacteristic
189C4⋊Q8C22×C2311yescharacteristic
190#126—21no—
191#126—21no—
192#126—21no—
193#125—21no—
194#125—21no—
195#125C22×C2311yescharacteristic
196#124—21no—
197#124C22×C2311yescharacteristic
198#123—21no—
199C22⋊Q8—21no—
200C22⋊Q8C22×C2311yescharacteristic
201C22⋊Q8C22×C2311yescharacteristic
202C22⋊Q8C22×C2311yescharacteristic
203#121C22×C2311yescharacteristic
204C4×Q8C22×C2311yescharacteristic
205C4×Q8C22×C2311yescharacteristic
206C4×Q8C22×C2311yescharacteristic
207C4×D4C22×C2311yescharacteristic
208#117C22×C2311yescharacteristic
209#107C22×C2311yescharacteristic
210#107C22×C2311yescharacteristic
211#106C22×C2311yescharacteristic
212#106C22×C2311yescharacteristic
213#103C22×C2311yescharacteristic
214#103C22×C2311yescharacteristic
215#103C22×C2311yescharacteristic
216#103C22×C2311yescharacteristic
217#103C22×C2311yescharacteristic
218#103C22×C2311yescharacteristic
219#102C22×C2311yescharacteristic
220#102C22×C2311yescharacteristic
221#97C22×C2311yescharacteristic
222#97C22×C2311yescharacteristic
223C4×C8C22×C2311yescharacteristic
224C4×C8C22×C2311yescharacteristic
225C23#317811yescharacteristic, direct product factor, Sylow
226C2×Q8—21no—
227C2×Q8C23×C2311yescharacteristic
228C2×Q8C23×C2311yescharacteristic
229C2×Q8C23×C2311yescharacteristic
230C2×D4—21no—
231C2×D4C23×C2311yescharacteristic
232C22×C4—21no—
233C22×C4C23×C2311yescharacteristic
234Q16—22no—
235Q16—22no—
236SD16—22no—
237SD16—22no—
238M16—22no—
239C2×C8—22no—
240C2×C8C23×C2311yescharacteristic
241C2×C8C23×C2311yescharacteristic
242C2×C8C23×C2311yescharacteristic
243C2×C8C23×C2311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4D4×C2311yescharacteristic
256C4⋊C4D4×C2311yescharacteristic
257C4⋊C4C23×C2311yescharacteristic
258C4⋊C4C23×C2311yescharacteristic
259C4⋊C4C23×C2311yescharacteristic
260C4⋊C4C23×C2311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4D4×C2311yescharacteristic
269C22⋊C4D4×C2311yescharacteristic
270C42—21no—
271C42—21no—
272C42—21no—
273C42C23×C2311yescharacteristic
274C42C23×C2311yescharacteristic
275C23—21no—
276C23C2×D4×C2311yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—81no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—81no—
297C2×C4C2×D4×C2311yescharacteristic
298C2×C4C2×D4×C2311yescharacteristic
299C2×C4C2×D4×C2311yescharacteristic
300C2×C4C2×D4×C2311yescharacteristic
301C2×C4C2×D4×C2311yescharacteristic
302C2×C4C24×C2311yescharacteristic
303C8—21no—
304C8—21no—
305C8#41×C2312yes—
306C8#41×C2312yes—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C22×D4×C2311yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4C2×#41×C2311yescharacteristic
327C4C2×#41×C2311yescharacteristic
328C2—41no—
329C2—81no—
330C2C23×#53211yescharacteristic
331C2C23×#57711yescharacteristic
332C2C23×#57811yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#541C2semidirect
3. C23×#501C2semidirect
4. C23×#497C2semidirect
5. C23×#488C2semidirect
6. C23×#487C2semidirect
7. C23×#470C2semidirect
9. C23×#467C2semidirect
11. C23×#441C2semidirect
12. C23×#440C2semidirect
13. C4×Q16×C23C2semidirect
14. C4×SD16×C23C2semidirect
15. C23×#405C2semidirect
140. #3178C23direct (N × H)
163. #541C2×C23semidirect
164. #501C2×C23semidirect
165. #497C2×C23semidirect
166. #488C2×C23semidirect
167. #487C2×C23semidirect
168. #470C2×C23semidirect
170. #467C2×C23semidirect
172. #441C2×C23semidirect
173. #440C2×C23semidirect
174. C4×Q16C2×C23semidirect
175. C4×SD16C2×C23semidirect
176. #405C2×C23semidirect
225. C23#3178direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5422
C23×#5412
C23×#5012
C23×#4972
C23×#4882
C23×#4872
C23×#4702
C23×#4682
C23×#4672
C23×#4652
C23×#4412
C23×#4402
C4×Q16×C232
C4×SD16×C232
C23×#4052
#317823