Finite Group Databasebeta

← C23×#3371 #15670990

Subgroups — C23×#3371 #15670990

Order 2944.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#559C211yesmaximal, characteristic
2C23×#557C211yesmaximal, characteristic, complemented
3C23×#490C211yesmaximal, characteristic
4C23×#489C211yesmaximal, characteristic, complemented
5C23×#488C211yesmaximal, characteristic, complemented
6C23×#479C211yesmaximal, characteristic, complemented
7C23×#478C211yesmaximal, characteristic, complemented
8C23×#476C211yesmaximal, characteristic, complemented
9C23×#464C211yesmaximal, characteristic, complemented
10C23×#464C211yesmaximal, characteristic, complemented
11C23×#460C211yesmaximal, characteristic
12C23×#446C211yesmaximal, characteristic, complemented
13C23×#441C211yesmaximal, characteristic, complemented
14C23×#440C211yesmaximal, characteristic, complemented
15C4×SD16×C23C211yesmaximal, characteristic, complemented
16C2×Q16×C23C2211yescharacteristic
17C2×SD16×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C23×C4⋊Q8—21no—
20C23×C4⋊Q8C2211yescharacteristic
21C23×#127C2211yescharacteristic
22C23×#125—21no—
23C23×#125—21no—
24C23×#125C2211yescharacteristic
25C23×#125C2211yescharacteristic
26C23×#124—21no—
27C23×#124C2211yescharacteristic
28C23×#124C2211yescharacteristic
29C23×#123—21no—
30C23×#123—21no—
31C4×Q8×C23—21no—
32C4×Q8×C23C2211yescharacteristic
33C4×Q8×C23C2211yescharacteristic
34C4×Q8×C23C2211yescharacteristic
35C4×Q8×C23C2211yescharacteristic
36C4×Q8×C23C2211yescharacteristic
37C4×D4×C23—21no—
38C4×D4×C23C2211yescharacteristic
39C4×D4×C23C2211yescharacteristic
40C23×#107C2211yescharacteristic
41C23×#106C2211yescharacteristic
42C23×#106C2211yescharacteristic
43C23×C4⋊C8C2211yescharacteristic
44C23×C4⋊C8C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×#103C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×#97C2211yescharacteristic
57C23×#97C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59C2×Q8×C23—21no—
60C2×Q8×C23D411yescharacteristic
61C2×Q8×C23C2311yescharacteristic
62C2×Q8×C23C2311yescharacteristic
63C2×D4×C23—21no—
64C2×D4×C23—21no—
65C2×D4×C23C2311yescharacteristic
66C2×D4×C23C2311yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69Q16×C23—22no—
70SD16×C23—22no—
71SD16×C23—22no—
72C2×C8×C23C2311yescharacteristic
73C2×C8×C23C2311yescharacteristic
74C2×C8×C23C2311yescharacteristic
75C2×C8×C23C2311yescharacteristic
76C4⋊C4×C23—21no—
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×C23D411yescharacteristic
86C4⋊C4×C23D411yescharacteristic
87C4⋊C4×C23D411yescharacteristic
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2311yescharacteristic
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C22⋊C4×C23—21no—
93C22⋊C4×C23—21no—
94C22⋊C4×C23—21no—
95C22⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C42×C23—21no—
99C42×C23—21no—
100C42×C23—21no—
101C42×C23—21no—
102C42×C23C2311yescharacteristic
103C42×C23C2311yescharacteristic
104C42×C23C2311yescharacteristic
105C23×C23—21no—
106C23×C23—21no—
107Q8×C23—21no—
108Q8×C23—21no—
109Q8×C23—21no—
110Q8×C23—41no—
111Q8×C23—42no—
112Q8×C23#4112yes—
113D4×C23—21no—
114D4×C23—21no—
115D4×C23—21no—
116D4×C23—21no—
117D4×C23—42no—
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—81no—
126C2×C4×C23—81no—
127C2×C4×C23C2×D411yescharacteristic
128C2×C4×C23C2×D411yescharacteristic
129C2×C4×C23C2×D411yescharacteristic
130C2×C4×C23C2×D411yescharacteristic
131C2×C4×C23C2×D411yescharacteristic
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2411yescharacteristic
134C8×C23—21no—
135C8×C23—21no—
136C8×C23—21no—
137C8×C23—21no—
138#3371C2311yesmaximal, characteristic, direct product factor, Sylow
139C22×C23—41no—
140C22×C23—41no—
141C22×C23—41no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23C22×D411yescharacteristic
146C4×C23—22no—
147C4×C23—21no—
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×C23C2×#4111yescharacteristic
159C4×C232−1+411yescharacteristic
160#559C2×C2311yescharacteristic
161#557C2×C2311yescharacteristic, complemented
162#490C2×C2311yescharacteristic
163#489C2×C2311yescharacteristic, complemented
164#488C2×C2311yescharacteristic, complemented
165#479C2×C2311yescharacteristic, complemented
166#478C2×C2311yescharacteristic, complemented
167#476C2×C2311yescharacteristic, complemented
168#464C2×C2311yescharacteristic, complemented
169#464C2×C2311yescharacteristic, complemented
170#460C2×C2311yescharacteristic
171#446C2×C2311yescharacteristic, complemented
172#441C2×C2311yescharacteristic, complemented
173#440C2×C2311yescharacteristic, complemented
174C4×SD16C2×C2311yescharacteristic, complemented
175C2×C23—81no—
176C2×C23—81no—
177C2×C23#54811yescharacteristic
178C2×C23#57511yescharacteristic
179C2×C23#57711yescharacteristic
180C2×Q16C22×C2311yescharacteristic
181C2×SD16C22×C2311yescharacteristic
182C2×SD16C22×C2311yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8C22×C2311yescharacteristic
185#127C22×C2311yescharacteristic
186#125—21no—
187#125—21no—
188#125C22×C2311yescharacteristic
189#125C22×C2311yescharacteristic
190#124—21no—
191#124C22×C2311yescharacteristic
192#124C22×C2311yescharacteristic
193#123—21no—
194#123—21no—
195C4×Q8—21no—
196C4×Q8C22×C2311yescharacteristic
197C4×Q8C22×C2311yescharacteristic
198C4×Q8C22×C2311yescharacteristic
199C4×Q8C22×C2311yescharacteristic
200C4×Q8C22×C2311yescharacteristic
201C4×D4—21no—
202C4×D4C22×C2311yescharacteristic
203C4×D4C22×C2311yescharacteristic
204#107C22×C2311yescharacteristic
205#106C22×C2311yescharacteristic
206#106C22×C2311yescharacteristic
207C4⋊C8C22×C2311yescharacteristic
208C4⋊C8C22×C2311yescharacteristic
209C4⋊C8C22×C2311yescharacteristic
210#103C22×C2311yescharacteristic
211#103C22×C2311yescharacteristic
212#103C22×C2311yescharacteristic
213#103C22×C2311yescharacteristic
214#102C22×C2311yescharacteristic
215#102C22×C2311yescharacteristic
216#102C22×C2311yescharacteristic
217#102C22×C2311yescharacteristic
218#102C22×C2311yescharacteristic
219#102C22×C2311yescharacteristic
220#97C22×C2311yescharacteristic
221#97C22×C2311yescharacteristic
222C4×C8C22×C2311yescharacteristic
223C23#337111yescharacteristic, direct product factor, Sylow
224C2×Q8—21no—
225C2×Q8D4×C2311yescharacteristic
226C2×Q8C23×C2311yescharacteristic
227C2×Q8C23×C2311yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C23×C2311yescharacteristic
231C2×D4C23×C2311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234Q16—22no—
235SD16—22no—
236SD16—22no—
237C2×C8C23×C2311yescharacteristic
238C2×C8C23×C2311yescharacteristic
239C2×C8C23×C2311yescharacteristic
240C2×C8C23×C2311yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4D4×C2311yescharacteristic
251C4⋊C4D4×C2311yescharacteristic
252C4⋊C4D4×C2311yescharacteristic
253C4⋊C4C23×C2311yescharacteristic
254C4⋊C4C23×C2311yescharacteristic
255C4⋊C4C23×C2311yescharacteristic
256C4⋊C4C23×C2311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42C23×C2311yescharacteristic
268C42C23×C2311yescharacteristic
269C42C23×C2311yescharacteristic
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—41no—
276Q8—42no—
277Q8#41×C2312yes—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×D4×C2311yescharacteristic
293C2×C4C2×D4×C2311yescharacteristic
294C2×C4C2×D4×C2311yescharacteristic
295C2×C4C2×D4×C2311yescharacteristic
296C2×C4C2×D4×C2311yescharacteristic
297C2×C4C2×D4×C2311yescharacteristic
298C2×C4C24×C2311yescharacteristic
299C8—21no—
300C8—21no—
301C8—21no—
302C8—21no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C22×D4×C2311yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4C2×#41×C2311yescharacteristic
323C4C23×2−1+411yescharacteristic
324C2—81no—
325C2—81no—
326C2C23×#54811yescharacteristic
327C2C23×#57511yescharacteristic
328C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#557C2semidirect
4. C23×#489C2semidirect
5. C23×#488C2semidirect
6. C23×#479C2semidirect
7. C23×#478C2semidirect
8. C23×#476C2semidirect
9. C23×#464C2semidirect
10. C23×#464C2semidirect
12. C23×#446C2semidirect
13. C23×#441C2semidirect
14. C23×#440C2semidirect
15. C4×SD16×C23C2semidirect
138. #3371C23direct (N × H)
161. #557C2×C23semidirect
163. #489C2×C23semidirect
164. #488C2×C23semidirect
165. #479C2×C23semidirect
166. #478C2×C23semidirect
167. #476C2×C23semidirect
168. #464C2×C23semidirect
169. #464C2×C23semidirect
171. #446C2×C23semidirect
172. #441C2×C23semidirect
173. #440C2×C23semidirect
174. C4×SD16C2×C23semidirect
223. C23#3371direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5592
C23×#5572
C23×#4902
C23×#4892
C23×#4882
C23×#4792
C23×#4782
C23×#4762
C23×#4642
C23×#4602
C23×#4462
C23×#4412
C23×#4402
C4×SD16×C232
#337123