Finite Group Databasebeta

← C23×#33866 #19181492

Subgroups — C23×#33866 #19181492

Order 5888.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3338C211yesmaximal, characteristic
2D4×Q16×C23C211yesmaximal, characteristic
3C23×#2241C211yesmaximal, characteristic, complemented
4C23×#2236C212yesmaximal, complemented
5C23×#2227C211yesmaximal, characteristic, complemented
6C23×#2216C212yesmaximal, complemented
7C23×#2209C211yesmaximal, characteristic, complemented
8C23×#2188C212yesmaximal, complemented
9C4×C23×Q32C211yesmaximal, characteristic, complemented
10D4×C16×C23C211yesmaximal, characteristic
11C2×C23×#358C212yesmaximal, complemented
12C22×Q16×C23C2212yes—
13C23×#554—21no—
14D4×Q8×C23—21no—
15C2×C23×Q32C2211yescharacteristic
16C2×C23×Q32C2212yes—
17C22×C16×C23C2212yes—
18C23×#500C2211yescharacteristic
19C23×#494C2211yescharacteristic
20C23×#484—22no—
21C23×#477—21no—
22C23×#467C2212yes—
23C23×#467C2212yes—
24C23×#462—21no—
25C23×#451—22no—
26C4×Q16×C23C2211yescharacteristic
27C4×Q16×C23C2211yescharacteristic
28C8×D4×C23C2211yescharacteristic
29C2×C23×#107C2212yes—
30C23×#366C2211yescharacteristic
31C23×#366C2212yes—
32C23×C4⋊C16C2211yescharacteristic
33C23×#358C2211yescharacteristic
34C23×#358C2211yescharacteristic
35C23×#358C2212yes—
36C23×#358C2212yes—
37C23×#358C2212yes—
38C23×#358C2212yes—
39C23×C22⋊C16C2212yes—
40C4×C16×C23C2211yescharacteristic
41C22×Q8×C23—22no—
42C2×Q16×C23—24no—
43C2×Q16×C23—22no—
44C2×Q16×C23—21no—
45C2×Q16×C23—42no—
46C2×Q16×C23C2312yes—
47C2×Q16×C23C2311yescharacteristic
48C2×Q16×C23C2311yescharacteristic
49C22×C8×C23C2312yes—
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8—21no—
52C23×C4⋊Q8—41no—
53C23×#125—41no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—42no—
57C23×C22⋊Q8—42no—
58C4×Q8×C23—21no—
59C4×Q8×C23—21no—
60C4×D4×C23—41no—
61C4×D4×C23—41no—
62C4×D4×C23D411yescharacteristic
63C2×C4⋊C4×C23—22no—
64C23×Q32—22no—
65C23×Q32—42no—
66C2×C16×C23—42no—
67C2×C16×C23C2311yescharacteristic
68C2×C16×C23C2311yescharacteristic
69C2×C16×C23C2312yes—
70C23×#107—22no—
71C23×#107—21no—
72C23×#107C2311yescharacteristic
73C23×#107C2312yes—
74C23×#107C2311yescharacteristic
75C23×C4⋊C8D411yescharacteristic
76C23×#103—21no—
77C23×#103—21no—
78C23×#103—22no—
79C23×#103—22no—
80C23×C22⋊C8D412yes—
81C4×C8×C23C2311yescharacteristic
82C2×Q8×C23—21no—
83C2×Q8×C23—21no—
84C2×Q8×C23—22no—
85C2×Q8×C23—44no—
86C2×Q8×C23—42no—
87C2×Q8×C23—41no—
88C2×Q8×C23—82no—
89C2×D4×C23D811yescharacteristic
90C22×C4×C23—42no—
91C22×C4×C23—42no—
92C22×C4×C23C2×D412yes—
93Q16×C23—21no—
94Q16×C23—22no—
95Q16×C23—21no—
96Q16×C23—21no—
97Q16×C23—22no—
98Q16×C23—44no—
99Q16×C23—81no—
100Q16×C23#4112yes—
101C2×C8×C23—22no—
102C2×C8×C23—22no—
103C2×C8×C23C2×D412yes—
104C2×C8×C23C2×D411yescharacteristic
105C2×C8×C23C2411yescharacteristic
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—42no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—42no—
115C4⋊C4×C23—42no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23D811yescharacteristic
118C22⋊C4×C23—42no—
119C22⋊C4×C23—42no—
120C22⋊C4×C23D812yes—
121C42×C23—41no—
122C42×C23—41no—
123C42×C23C2×D411yescharacteristic
124C16×C23—21no—
125C16×C23—21no—
126C16×C23—22no—
127#33866C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23C2×D812yes—
129Q8×C23—22no—
130Q8×C23—41no—
131Q8×C23—42no—
132Q8×C23—41no—
133Q8×C23—42no—
134Q8×C23—41no—
135Q8×C23—84no—
136Q8×C23—161no—
137D4×C23Q3214yescomplemented
138C2×C4×C23—22no—
139C2×C4×C23—22no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—42no—
145C2×C4×C23—42no—
146C2×C4×C23—84no—
147C2×C4×C23—162no—
148C2×C4×C23C2×D811yescharacteristic
149C2×C4×C23C2×D812yes—
150C2×C4×C23C2×D811yescharacteristic
151C2×C4×C23C22×D411yescharacteristic
152C8×C23—22no—
153C8×C23—21no—
154C8×C23C2×#4111yescharacteristic
155C8×C23H(5,𝔽2)11yescharacteristic
156#3338C2×C2311yescharacteristic
157D4×Q16C2×C2311yescharacteristic
158#2241C2×C2311yescharacteristic, complemented
159#2236C2×C2312yescomplemented
160#2227C2×C2311yescharacteristic, complemented
161#2216C2×C2312yescomplemented
162#2209C2×C2311yescharacteristic, complemented
163#2188C2×C2312yescomplemented
164C4×Q32C2×C2311yescharacteristic, complemented
165D4×C16C2×C2311yescharacteristic
166C2×#358C2×C2312yescomplemented
167C22×C23—22no—
168C22×C23—22no—
169C22×C23C2×Q3214yes—
170C22×C23C22×D811yescharacteristic
171C4×C23—21no—
172C4×C23—22no—
173C4×C23—42no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23—82no—
177C4×C23—82no—
178C4×C23—81no—
179C4×C23C2×Q3212yes—
180C4×C23#54611yescharacteristic
181C4×C23#57711yescharacteristic
182C22×Q16C22×C2312yes—
183#554—21no—
184D4×Q8—21no—
185C2×Q32C22×C2311yescharacteristic
186C2×Q32C22×C2312yes—
187C22×C16C22×C2312yes—
188#500C22×C2311yescharacteristic
189#494C22×C2311yescharacteristic
190#484—22no—
191#477—21no—
192#467C22×C2312yes—
193#467C22×C2312yes—
194#462—21no—
195#451—22no—
196C4×Q16C22×C2311yescharacteristic
197C4×Q16C22×C2311yescharacteristic
198C8×D4C22×C2311yescharacteristic
199C2×#107C22×C2312yes—
200#366C22×C2311yescharacteristic
201#366C22×C2312yes—
202C4⋊C16C22×C2311yescharacteristic
203#358C22×C2311yescharacteristic
204#358C22×C2312yes—
205#358C22×C2312yes—
206#358C22×C2312yes—
207#358C22×C2311yescharacteristic
208#358C22×C2312yes—
209C22⋊C16C22×C2312yes—
210C4×C16C22×C2311yescharacteristic
211C2×C23—24no—
212C2×C23#329411yescharacteristic
213C2×C23C22×Q3211yescharacteristic
214C2×C23#341611yescharacteristic
215C22×Q8—22no—
216C2×Q16—24no—
217C2×Q16—22no—
218C2×Q16—21no—
219C2×Q16—42no—
220C2×Q16C23×C2312yes—
221C2×Q16C23×C2311yescharacteristic
222C2×Q16C23×C2311yescharacteristic
223C22×C8C23×C2312yes—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—41no—
227#125—41no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4D4×C2311yescharacteristic
237C2×C4⋊C4—22no—
238Q32—22no—
239Q32—42no—
240C2×C16—42no—
241C2×C16C23×C2311yescharacteristic
242C2×C16C23×C2311yescharacteristic
243C2×C16C23×C2312yes—
244#107—22no—
245#107—21no—
246#107C23×C2311yescharacteristic
247#107C23×C2312yes—
248#107C23×C2311yescharacteristic
249C4⋊C8D4×C2311yescharacteristic
250#103—21no—
251#103—21no—
252#103—22no—
253#103—22no—
254C22⋊C8D4×C2312yes—
255C4×C8C23×C2311yescharacteristic
256C23#3386611yescharacteristic, direct product factor, Sylow
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—82no—
264C2×D4D8×C2311yescharacteristic
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×D4×C2312yes—
268Q16—21no—
269Q16—22no—
270Q16—21no—
271Q16—21no—
272Q16—22no—
273Q16—44no—
274Q16—81no—
275Q16#41×C2312yes—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×D4×C2311yescharacteristic
279C2×C8C2×D4×C2312yes—
280C2×C8C24×C2311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4D8×C2311yescharacteristic
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4D8×C2312yes—
296C42—41no—
297C42—41no—
298C42C2×D4×C2311yescharacteristic
299C16—21no—
300C16—21no—
301C16—22no—
302C23C2×D8×C2312yes—
303Q8—22no—
304Q8—41no—
305Q8—42no—
306Q8—41no—
307Q8—42no—
308Q8—41no—
309Q8—84no—
310Q8—161no—
311D4C23×Q3214yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—162no—
322C2×C4C2×D8×C2311yescharacteristic
323C2×C4C2×D8×C2312yes—
324C2×C4C2×D8×C2311yescharacteristic
325C2×C4C22×D4×C2311yescharacteristic
326C8—21no—
327C8—22no—
328C8C2×#41×C2311yescharacteristic
329C8C23×H(5,𝔽2)11yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×C23×Q3214yes—
333C22C22×D8×C2311yescharacteristic
334C4—22no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—81no—
340C4—82no—
341C4—81no—
342C4C2×C23×Q3212yes—
343C4C23×#54611yescharacteristic
344C4C23×#57711yescharacteristic
345C2—24no—
346C2C23×#329411yescharacteristic
347C2C22×C23×Q3211yescharacteristic
348C2C23×#341611yescharacteristic
subgroupG/H
center, socleC22×C23C22×D8
commutator, Frattini, cosocleC2×C8C24×C23
prime radicalC2×D4×C23D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×D4×C23 ◁ C8×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#2241C2semidirect
4. C23×#2236C2semidirect
5. C23×#2227C2semidirect
6. C23×#2216C2semidirect
7. C23×#2209C2semidirect
8. C23×#2188C2semidirect
9. C4×C23×Q32C2semidirect
11. C2×C23×#358C2semidirect
127. #33866C23direct (N × H)
137. D4×C23Q32semidirect
158. #2241C2×C23semidirect
159. #2236C2×C23semidirect
160. #2227C2×C23semidirect
161. #2216C2×C23semidirect
162. #2209C2×C23semidirect
163. #2188C2×C23semidirect
164. C4×Q32C2×C23semidirect
166. C2×#358C2×C23semidirect
256. C23#33866direct (N × H)
311. D4C23×Q32semidirect

Maximal subgroups

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

subgroupindex
C23×#33382
D4×Q16×C232
C23×#22412
C23×#22362
C23×#22272
C23×#22162
C23×#22092
C23×#21882
C4×C23×Q322
D4×C16×C232
C2×C23×#3582
#3386623