Finite Group Databasebeta

← C23×#34978 #19182604

Subgroups — C23×#34978 #19182604

Order 5888.

1834 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2612C214yesmaximal, complemented
2C23×#2571C211yesmaximal, characteristic, complemented
3C23×#2552C212yesmaximal, complemented
4C23×#2551C211yesmaximal, characteristic, complemented
5C23×#2441C214yesmaximal
6C23×#2439C214yesmaximal
7C23×#2396C214yesmaximal
8C4×C23×C4⋊Q8C214yesmaximal, complemented
9C4×C23×#125C214yesmaximal, complemented
10C23×#2285C211yesmaximal, characteristic
11C4×C23×#117C212yesmaximal
12C2×C23×C4⋊Q8C2214yes—
13C2×C23×#125C2214yes—
14C2×C4×Q8×C23C2214yes—
15C2×C23×#117C2212yes—
16C2×C23×#117C2212yes—
17C2×C23×#117C2212yes—
18C22×C42×C23C2211yescharacteristic
19C23×#400C2214yes—
20C23×#399C2214yes—
21C23×#398C2218yes—
22C23×#396C2218yes—
23C23×#395C2214yes—
24C23×#393C2214yes—
25C23×#391C2218yes—
26C23×#389C2214yes—
27C23×#389C2214yes—
28C23×#389C2214yes—
29C23×#389C2214yes—
30C23×#389C2218yes—
31C23×#387C2218yes—
32C23×#387C2218yes—
33C23×#384C2214yes—
34C23×#384C2211yescharacteristic
35C23×#384C2211yescharacteristic
36C23×#382C2214yes—
37C23×#382C2212yes—
38C23×#380C2214yes—
39C4×C4⋊C4×C23C2212yes—
40C4×C4⋊C4×C23C2212yes—
41C4×C4⋊C4×C23C2212yes—
42C4×C4⋊C4×C23C2212yes—
43C4×C4⋊C4×C23C2214yes—
44C4×C4⋊C4×C23C2214yes—
45C4×C4⋊C4×C23C2218yes—
46C4×C22⋊C4×C23C2214yes—
47C4×C22⋊C4×C23C2214yes—
48C23×#376C2214yes—
49C43×C23C2214yes—
50C22×Q8×C23C2314yes—
51C23×C4×C23C2312yes—
52C23×C4×C23C2311yescharacteristic
53C23×C4⋊Q8—216no—
54C23×#125—216no—
55C4×Q8×C23—48no—
56C23×#117—28no—
57C23×#117—28no—
58C23×#117—44no—
59C2×C4⋊C4×C23C2312yes—
60C2×C4⋊C4×C23C2312yes—
61C2×C4⋊C4×C23C2314yes—
62C2×C4⋊C4×C23C2314yes—
63C2×C4⋊C4×C23C2318yes—
64C2×C4⋊C4×C23C2318yes—
65C2×C4⋊C4×C23C2318yes—
66C2×C4⋊C4×C23C2312yes—
67C2×C4⋊C4×C23C2312yes—
68C2×C4⋊C4×C23C2318yes—
69C2×C4⋊C4×C23C2314yes—
70C2×C4⋊C4×C23C2312yes—
71C2×C4⋊C4×C23C2312yes—
72C2×C22⋊C4×C23C2314yes—
73C2×C22⋊C4×C23C2314yes—
74C2×C22⋊C4×C23C2314yes—
75C2×C42×C23Q818yescomplemented
76C2×C42×C23C2314yes—
77C2×C42×C23C2314yes—
78C2×C42×C23C2314yes—
79C2×C42×C23C2314yes—
80C2×C42×C23C2314yes—
81C2×C42×C23C2314yes—
82C2×C42×C23C2314yes—
83C2×C42×C23C2311yescharacteristic
84C2×C42×C23C2311yescharacteristic
85C2×C42×C23C2312yes—
86C23×#95C2314yes—
87C23×#95C2314yes—
88C23×#95C2314yes—
89C23×#95C2314yes—
90C23×#95C2318yes—
91C23×#95C2314yes—
92C23×#95C2318yes—
93C23×#95C2314yes—
94C23×#95C2318yes—
95C24×C23C2411yescharacteristic
96C2×Q8×C23—216no—
97C2×Q8×C23—44no—
98C22×C4×C23—24no—
99C22×C4×C23—24no—
100C22×C4×C23—22no—
101C22×C4×C23—22no—
102C22×C4×C23C2×Q818yes—
103C22×C4×C23C2×Q814yes—
104C22×C4×C23C2414yes—
105C22×C4×C23C2414yes—
106C22×C4×C23C2414yes—
107C22×C4×C23C2414yes—
108C22×C4×C23C2412yes—
109C22×C4×C23C2411yescharacteristic
110C22×C4×C23C2414yes—
111C22×C4×C23C2414yes—
112C22×C4×C23C2412yes—
113C22×C4×C23C2411yescharacteristic
114C4⋊C4×C23—24no—
115C4⋊C4×C23—24no—
116C4⋊C4×C23—216no—
117C4⋊C4×C23—216no—
118C4⋊C4×C23—28no—
119C4⋊C4×C23—216no—
120C4⋊C4×C23—28no—
121C4⋊C4×C23—42no—
122C4⋊C4×C23—42no—
123C4⋊C4×C23—48no—
124C4⋊C4×C23—44no—
125C4⋊C4×C23#4118yes—
126C4⋊C4×C23#4118yes—
127C22⋊C4×C23—28no—
128C22⋊C4×C23—28no—
129C22⋊C4×C23—44no—
130C42×C23—28no—
131C42×C23—28no—
132C42×C23—28no—
133C42×C23—28no—
134C42×C23—28no—
135C42×C23—28no—
136C42×C23—28no—
137C42×C23C2×Q814yes—
138C42×C23C2×Q814yes—
139C42×C23C2×Q818yes—
140#34978C2311yesmaximal, characteristic, direct product factor, Sylow
141C23×C23—22no—
142C23×C23—22no—
143C23×C23—21no—
144C23×C23—21no—
145C23×C23C22×Q812yes—
146C23×C23C2511yescharacteristic
147Q8×C23—416no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—24no—
151C2×C4×C23—24no—
152C2×C4×C23—24no—
153C2×C4×C23—21no—
154C2×C4×C23—24no—
155C2×C4×C23—21no—
156C2×C4×C23—216no—
157C2×C4×C23—28no—
158C2×C4×C23—24no—
159C2×C4×C23—24no—
160C2×C4×C23—24no—
161C2×C4×C23—24no—
162C2×C4×C23—28no—
163C2×C4×C23—28no—
164C2×C4×C23—24no—
165C2×C4×C23—24no—
166C2×C4×C23—24no—
167C2×C4×C23—24no—
168C2×C4×C23C22×Q814yes—
169C2×C4×C23C22×Q812yes—
170C2×C4×C23C22×Q814yes—
171C2×C4×C23C22×Q812yes—
172C2×C4×C23C2×#4118yes—
173C2×C4×C23C2×#4118yes—
174C2×C4×C23C2×#4114yes—
175C2×C4×C23C2×#4114yes—
176C2×C4×C23H(5,𝔽2)12yes—
177C2×C4×C232−1+412yes—
178#2612C2×C2314yescomplemented
179#2571C2×C2311yescharacteristic, complemented
180#2552C2×C2312yescomplemented
181#2551C2×C2311yescharacteristic, complemented
182#2441C2×C2314yes—
183#2439C2×C2314yes—
184#2396C2×C2314yes—
185C4×C4⋊Q8C2×C2314yescomplemented
186C4×#125C2×C2314yescomplemented
187#2285C2×C2311yescharacteristic
188C4×#117C2×C2312yes—
189C22×C23—24no—
190C22×C23—22no—
191C22×C23—41no—
192C22×C23—42no—
193C22×C23—41no—
194C22×C23C23×Q811yescharacteristic
195C22×C23C22×#4112yes—
196C22×C23C22×#4112yes—
197C22×C23C2×H(5,𝔽2)11yescharacteristic
198C22×C23C2×2−1+411yescharacteristic
199C4×C23—24no—
200C4×C23—22no—
201C4×C23—28no—
202C4×C23—28no—
203C4×C23—44no—
204C4×C23—44no—
205C4×C23—44no—
206C4×C23—44no—
207C4×C23#53318yes—
208C4×C23#54414yes—
209C2×C4⋊Q8C22×C2314yes—
210C2×#125C22×C2314yes—
211C2×C4×Q8C22×C2314yes—
212C2×#117C22×C2312yes—
213C2×#117C22×C2312yes—
214C2×#117C22×C2312yes—
215C22×C42C22×C2311yescharacteristic
216#400C22×C2314yes—
217#399C22×C2314yes—
218#398C22×C2318yes—
219#396C22×C2318yes—
220#395C22×C2314yes—
221#393C22×C2314yes—
222#391C22×C2318yes—
223#389C22×C2314yes—
224#389C22×C2314yes—
225#389C22×C2314yes—
226#389C22×C2318yes—
227#389C22×C2314yes—
228#387C22×C2318yes—
229#387C22×C2318yes—
230#384C22×C2314yes—
231#384C22×C2311yescharacteristic
232#384C22×C2311yescharacteristic
233#382C22×C2314yes—
234#382C22×C2312yes—
235#380C22×C2314yes—
236C4×C4⋊C4C22×C2314yes—
237C4×C4⋊C4C22×C2312yes—
238C4×C4⋊C4C22×C2318yes—
239C4×C4⋊C4C22×C2314yes—
240C4×C4⋊C4C22×C2312yes—
241C4×C4⋊C4C22×C2312yes—
242C4×C4⋊C4C22×C2312yes—
243C4×C22⋊C4C22×C2314yes—
244C4×C22⋊C4C22×C2314yes—
245#376C22×C2314yes—
246C43C22×C2314yes—
247C2×C23—42no—
248C2×C23C2×#53312yes—
249C2×C23C2×#54411yescharacteristic
250C2×C23#348111yescharacteristic
251C2×C23#348212yes—
252C2×C23#348311yescharacteristic
253C22×Q8C23×C2314yes—
254C23×C4C23×C2311yescharacteristic
255C23×C4C23×C2312yes—
256C4⋊Q8—216no—
257#125—216no—
258C4×Q8—48no—
259#117—28no—
260#117—28no—
261#117—44no—
262C2×C4⋊C4C23×C2312yes—
263C2×C4⋊C4C23×C2312yes—
264C2×C4⋊C4C23×C2314yes—
265C2×C4⋊C4C23×C2318yes—
266C2×C4⋊C4C23×C2318yes—
267C2×C4⋊C4C23×C2314yes—
268C2×C4⋊C4C23×C2318yes—
269C2×C4⋊C4C23×C2312yes—
270C2×C4⋊C4C23×C2312yes—
271C2×C4⋊C4C23×C2318yes—
272C2×C4⋊C4C23×C2314yes—
273C2×C4⋊C4C23×C2312yes—
274C2×C4⋊C4C23×C2312yes—
275C2×C22⋊C4C23×C2314yes—
276C2×C22⋊C4C23×C2314yes—
277C2×C22⋊C4C23×C2314yes—
278C2×C42Q8×C2318yescomplemented
279C2×C42C23×C2314yes—
280C2×C42C23×C2314yes—
281C2×C42C23×C2314yes—
282C2×C42C23×C2314yes—
283C2×C42C23×C2314yes—
284C2×C42C23×C2314yes—
285C2×C42C23×C2314yes—
286C2×C42C23×C2312yes—
287C2×C42C23×C2311yescharacteristic
288C2×C42C23×C2311yescharacteristic
289#95C23×C2314yes—
290#95C23×C2314yes—
291#95C23×C2314yes—
292#95C23×C2314yes—
293#95C23×C2318yes—
294#95C23×C2314yes—
295#95C23×C2318yes—
296#95C23×C2314yes—
297#95C23×C2318yes—
298C23#3497811yescharacteristic, direct product factor, Sylow
299C24C24×C2311yescharacteristic
300C2×Q8—216no—
301C2×Q8—44no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4C2×Q8×C2318yes—
307C22×C4C2×Q8×C2314yes—
308C22×C4C24×C2314yes—
309C22×C4C24×C2314yes—
310C22×C4C24×C2314yes—
311C22×C4C24×C2314yes—
312C22×C4C24×C2312yes—
313C22×C4C24×C2311yescharacteristic
314C22×C4C24×C2314yes—
315C22×C4C24×C2314yes—
316C22×C4C24×C2312yes—
317C22×C4C24×C2311yescharacteristic
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—216no—
321C4⋊C4—216no—
322C4⋊C4—28no—
323C4⋊C4—216no—
324C4⋊C4—28no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C4⋊C4—44no—
329C4⋊C4#41×C2318yes—
330C4⋊C4#41×C2318yes—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C42—28no—
335C42—28no—
336C42—28no—
337C42—28no—
338C42—28no—
339C42—28no—
340C42—28no—
341C42C2×Q8×C2314yes—
342C42C2×Q8×C2314yes—
343C42C2×Q8×C2318yes—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23C22×Q8×C2312yes—
349C23C25×C2311yescharacteristic
350Q8—416no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—216no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4C22×Q8×C2314yes—
372C2×C4C22×Q8×C2312yes—
373C2×C4C22×Q8×C2314yes—
374C2×C4C22×Q8×C2312yes—
375C2×C4C2×#41×C2318yes—
376C2×C4C2×#41×C2318yes—
377C2×C4C2×#41×C2314yes—
378C2×C4C2×#41×C2314yes—
379C2×C4C23×H(5,𝔽2)12yes—
380C2×C4C23×2−1+412yes—
381C22—24no—
382C22—22no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22C23×Q8×C2311yescharacteristic
387C22C22×#41×C2312yes—
388C22C22×#41×C2312yes—
389C22C2×C23×H(5,𝔽2)11yescharacteristic
390C22C2×C23×2−1+411yescharacteristic
391C4—24no—
392C4—22no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4—44no—
399C4C23×#53318yes—
400C4C23×#54414yes—
401C2—42no—
402C2C2×C23×#53312yes—
403C2C2×C23×#54411yescharacteristic
404C2C23×#348111yescharacteristic
405C2C23×#348212yes—
406C2C23×#348311yescharacteristic
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2612C2semidirect
2. C23×#2571C2semidirect
3. C23×#2552C2semidirect
4. C23×#2551C2semidirect
8. C4×C23×C4⋊Q8C2semidirect
9. C4×C23×#125C2semidirect
75. C2×C42×C23Q8semidirect
140. #34978C23direct (N × H)
178. #2612C2×C23semidirect
179. #2571C2×C23semidirect
180. #2552C2×C23semidirect
181. #2551C2×C23semidirect
185. C4×C4⋊Q8C2×C23semidirect
186. C4×#125C2×C23semidirect
278. C2×C42Q8×C23semidirect
298. C23#34978direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26122
C23×#25712
C23×#25522
C23×#25512
C23×#24412
C23×#24392
C23×#23962
C4×C23×C4⋊Q82
C4×C23×#1252
C23×#22852
C4×C23×#1172
#3497823