Finite Group Databasebeta

← C23×#3269 #15670888

Subgroups — C23×#3269 #15670888

Order 2944.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#137C211yesmaximal, characteristic, complemented
2C2×C23×#135C211yesmaximal, characteristic, complemented
3C23×#548C211yesmaximal, characteristic, complemented
4C23×#547C211yesmaximal, characteristic, complemented
5C23×#497C211yesmaximal, characteristic, complemented
6C23×#468C211yesmaximal, characteristic, complemented
7C23×#467C211yesmaximal, characteristic, complemented
8C23×#463C211yesmaximal, characteristic, complemented
9C23×#462C211yesmaximal, characteristic, complemented
10C23×#452C211yesmaximal, characteristic, complemented
11C23×#452C211yesmaximal, characteristic, complemented
12C23×#451C211yesmaximal, characteristic, complemented
13C23×#449C211yesmaximal, characteristic, complemented
14C23×#440C211yesmaximal, characteristic, complemented
15C23×#435C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic
18C2×#41×C23C2211yescharacteristic
19C22×Q8×C23C2211yescharacteristic
20C23×#137—24no—
21C23×#135—24no—
22C2×Q16×C23C2211yescharacteristic
23C2×Q16×C23C2211yescharacteristic
24C2×Q16×C23C2211yescharacteristic
25C2×Q16×C23C2211yescharacteristic
26C2×SD16×C23C2211yescharacteristic
27C2×SD16×C23C2211yescharacteristic
28C2×SD16×C23C2211yescharacteristic
29C2×SD16×C23C2211yescharacteristic
30C2×D8×C23C2211yescharacteristic
31C2×M16×C23C2211yescharacteristic
32C22×C8×C23C2211yescharacteristic
33C23×C4⋊Q8C2211yescharacteristic
34C23×#124—21no—
35C23×#124C2211yescharacteristic
36C23×#123—21no—
37C23×#123—21no—
38C23×C22⋊Q8—21no—
39C23×C22⋊Q8C2211yescharacteristic
40C23×C22⋊Q8C2211yescharacteristic
41C23×C22⋊Q8C2211yescharacteristic
42C23×#121—21no—
43C23×#121—21no—
44C23×#121C2211yescharacteristic
45C4×Q8×C23C2211yescharacteristic
46C4×D4×C23—21no—
47C4×D4×C23C2211yescharacteristic
48C4×D4×C23C2211yescharacteristic
49C2×C4⋊C4×C23—21no—
50C23×#107C2211yescharacteristic
51C23×C4⋊C8C2211yescharacteristic
52C23×#103C2211yescharacteristic
53C23×#103C2211yescharacteristic
54C23×#103C2211yescharacteristic
55C23×#103C2211yescharacteristic
56C23×#102C2211yescharacteristic
57C23×#102C2211yescharacteristic
58C23×C22⋊C8C2211yescharacteristic
59C23×C22⋊C8C2211yescharacteristic
60C23×#97C2211yescharacteristic
61#41×C23—22no—
62#41×C23—22no—
63#41×C23—22no—
64#41×C23—41no—
65#41×C23—41no—
66#41×C23—41no—
67C2×Q8×C23—21no—
68C2×Q8×C23—22no—
69C2×Q8×C23—41no—
70C2×Q8×C23C2311yescharacteristic
71C2×Q8×C23C2311yescharacteristic
72C2×Q8×C23C2311yescharacteristic
73C2×Q8×C23C2311yescharacteristic
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23D411yescharacteristic
79C2×D4×C23C2311yescharacteristic
80C2×D4×C23C2311yescharacteristic
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23C2311yescharacteristic
87C22×C4×C23C2311yescharacteristic
88Q16×C23—22no—
89Q16×C23—22no—
90Q16×C23—22no—
91Q16×C23—41no—
92SD16×C23—22no—
93SD16×C23—22no—
94SD16×C23—41no—
95SD16×C23D414yescomplemented
96D8×C23—22no—
97M16×C23—22no—
98C2×C8×C23—22no—
99C2×C8×C23C2311yescharacteristic
100C2×C8×C23C2311yescharacteristic
101C2×C8×C23C2311yescharacteristic
102C2×C8×C23C2311yescharacteristic
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23D411yescharacteristic
110C4⋊C4×C23C2311yescharacteristic
111C4⋊C4×C23C2311yescharacteristic
112C22⋊C4×C23—21no—
113C22⋊C4×C23—21no—
114C22⋊C4×C23—21no—
115C22⋊C4×C23—21no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C22⋊C4×C23D411yescharacteristic
120C22⋊C4×C23D411yescharacteristic
121C42×C23—21no—
122C42×C23C2311yescharacteristic
123C23×C23—21no—
124C23×C23—21no—
125C23×C23C2×D411yescharacteristic
126C23×C23C2×D411yescharacteristic
127Q8×C23—21no—
128Q8×C23—21no—
129Q8×C23—21no—
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23—21no—
133Q8×C23—21no—
134Q8×C23—42no—
135Q8×C23C2×D412yes—
136D4×C23—21no—
137D4×C23—21no—
138D4×C23—21no—
139D4×C23—41no—
140D4×C23—42no—
141D4×C23—42no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23C2×D412yes—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—42no—
156C2×C4×C23—42no—
157C2×C4×C23—42no—
158C2×C4×C23—42no—
159C2×C4×C23—42no—
160C2×C4×C23C2×D411yescharacteristic
161C2×C4×C23C2×D411yescharacteristic
162C2×C4×C23C2×D411yescharacteristic
163C2×C4×C23C2×D411yescharacteristic
164C2×C4×C23C2411yescharacteristic
165C8×C23—21no—
166C8×C23—21no—
167C8×C23—21no—
168C8×C23C2×D412yes—
169#3269C2311yesmaximal, characteristic, direct product factor, Sylow
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—22no—
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23C22×D411yescharacteristic
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—22no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—41no—
192C4×C23—41no—
193C4×C23C22×D411yescharacteristic
194C4×C23H(5,𝔽2)11yescharacteristic
195C2×#137C2×C2311yescharacteristic, complemented
196C2×#135C2×C2311yescharacteristic, complemented
197#548C2×C2311yescharacteristic, complemented
198#547C2×C2311yescharacteristic, complemented
199#497C2×C2311yescharacteristic, complemented
200#468C2×C2311yescharacteristic, complemented
201#467C2×C2311yescharacteristic, complemented
202#463C2×C2311yescharacteristic, complemented
203#462C2×C2311yescharacteristic, complemented
204#452C2×C2311yescharacteristic, complemented
205#452C2×C2311yescharacteristic, complemented
206#451C2×C2311yescharacteristic, complemented
207#449C2×C2311yescharacteristic, complemented
208#440C2×C2311yescharacteristic, complemented
209#435C2×C2311yescharacteristic, complemented
210C2×C23—41no—
211C2×C23—41no—
212C2×C23—42no—
213C2×C23—81no—
214C2×C23D4211yescharacteristic
215C2×C23#57511yescharacteristic
216C2×C23#57811yescharacteristic
217C2×#41C22×C2311yescharacteristic
218C2×#41C22×C2311yescharacteristic
219C2×#41C22×C2311yescharacteristic
220C22×Q8C22×C2311yescharacteristic
221#137—24no—
222#135—24no—
223C2×Q16C22×C2311yescharacteristic
224C2×Q16C22×C2311yescharacteristic
225C2×Q16C22×C2311yescharacteristic
226C2×Q16C22×C2311yescharacteristic
227C2×SD16C22×C2311yescharacteristic
228C2×SD16C22×C2311yescharacteristic
229C2×SD16C22×C2311yescharacteristic
230C2×SD16C22×C2311yescharacteristic
231C2×D8C22×C2311yescharacteristic
232C2×M16C22×C2311yescharacteristic
233C22×C8C22×C2311yescharacteristic
234C4⋊Q8C22×C2311yescharacteristic
235#124—21no—
236#124C22×C2311yescharacteristic
237#123—21no—
238#123—21no—
239C22⋊Q8—21no—
240C22⋊Q8C22×C2311yescharacteristic
241C22⋊Q8C22×C2311yescharacteristic
242C22⋊Q8C22×C2311yescharacteristic
243#121—21no—
244#121—21no—
245#121C22×C2311yescharacteristic
246C4×Q8C22×C2311yescharacteristic
247C4×D4—21no—
248C4×D4C22×C2311yescharacteristic
249C4×D4C22×C2311yescharacteristic
250C2×C4⋊C4—21no—
251#107C22×C2311yescharacteristic
252C4⋊C8C22×C2311yescharacteristic
253#103C22×C2311yescharacteristic
254#103C22×C2311yescharacteristic
255#103C22×C2311yescharacteristic
256#103C22×C2311yescharacteristic
257#102C22×C2311yescharacteristic
258#102C22×C2311yescharacteristic
259C22⋊C8C22×C2311yescharacteristic
260C22⋊C8C22×C2311yescharacteristic
261#97C22×C2311yescharacteristic
262C23#326911yescharacteristic, direct product factor, Sylow
263#41—22no—
264#41—22no—
265#41—22no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8C23×C2311yescharacteristic
273C2×Q8C23×C2311yescharacteristic
274C2×Q8C23×C2311yescharacteristic
275C2×Q8C23×C2311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4D4×C2311yescharacteristic
281C2×D4C23×C2311yescharacteristic
282C2×D4C23×C2311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C23×C2311yescharacteristic
289C22×C4C23×C2311yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—22no—
293Q16—41no—
294SD16—22no—
295SD16—22no—
296SD16—41no—
297SD16D4×C2314yescomplemented
298D8—22no—
299M16—22no—
300C2×C8—22no—
301C2×C8C23×C2311yescharacteristic
302C2×C8C23×C2311yescharacteristic
303C2×C8C23×C2311yescharacteristic
304C2×C8C23×C2311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4D4×C2311yescharacteristic
312C4⋊C4C23×C2311yescharacteristic
313C4⋊C4C23×C2311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4D4×C2311yescharacteristic
322C22⋊C4D4×C2311yescharacteristic
323C42—21no—
324C42C23×C2311yescharacteristic
325C23—21no—
326C23—21no—
327C23C2×D4×C2311yescharacteristic
328C23C2×D4×C2311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8—42no—
337Q8C2×D4×C2312yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4C2×D4×C2312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4C2×D4×C2311yescharacteristic
363C2×C4C2×D4×C2311yescharacteristic
364C2×C4C2×D4×C2311yescharacteristic
365C2×C4C2×D4×C2311yescharacteristic
366C2×C4C24×C2311yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8C2×D4×C2312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C22×D4×C2311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4C22×D4×C2311yescharacteristic
395C4C23×H(5,𝔽2)11yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—81no—
400C2D42×C2311yescharacteristic
401C2C23×#57511yescharacteristic
402C2C23×#57811yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
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
1. C2×C23×#137C2semidirect
2. C2×C23×#135C2semidirect
3. C23×#548C2semidirect
4. C23×#547C2semidirect
5. C23×#497C2semidirect
6. C23×#468C2semidirect
7. C23×#467C2semidirect
8. C23×#463C2semidirect
9. C23×#462C2semidirect
10. C23×#452C2semidirect
11. C23×#452C2semidirect
12. C23×#451C2semidirect
13. C23×#449C2semidirect
14. C23×#440C2semidirect
15. C23×#435C2semidirect
95. SD16×C23D4semidirect
169. #3269C23direct (N × H)
195. C2×#137C2×C23semidirect
196. C2×#135C2×C23semidirect
197. #548C2×C23semidirect
198. #547C2×C23semidirect
199. #497C2×C23semidirect
200. #468C2×C23semidirect
201. #467C2×C23semidirect
202. #463C2×C23semidirect
203. #462C2×C23semidirect
204. #452C2×C23semidirect
205. #452C2×C23semidirect
206. #451C2×C23semidirect
207. #449C2×C23semidirect
208. #440C2×C23semidirect
209. #435C2×C23semidirect
262. C23#3269direct (N × H)
297. SD16D4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#1372
C2×C23×#1352
C23×#5482
C23×#5472
C23×#4972
C23×#4682
C23×#4672
C23×#4632
C23×#4622
C23×#4522
C23×#4512
C23×#4492
C23×#4402
C23×#4352
#326923