Finite Group Databasebeta

← C23×#3198 #15670817

Subgroups — C23×#3198 #15670817

Order 2944.

422 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×#537C211yesmaximal, characteristic, complemented
2C23×#535C211yesmaximal, characteristic, complemented
3C23×#485C211yesmaximal, characteristic, complemented
4C23×#482C211yesmaximal, characteristic, complemented
5C23×#481C211yesmaximal, characteristic, complemented
6C23×#480C211yesmaximal, characteristic, complemented
7C23×#469C211yesmaximal, characteristic, complemented
8C23×#468C211yesmaximal, characteristic, complemented
9C23×#466C211yesmaximal, characteristic, complemented
10C23×#465C211yesmaximal, characteristic, complemented
11C23×#450C211yesmaximal, characteristic, complemented
12C23×#449C211yesmaximal, characteristic, complemented
13C23×#449C211yesmaximal, characteristic, complemented
14C23×#447C211yesmaximal, characteristic, complemented
15C23×#408C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic, complemented
18C22×D4×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×SD16×C23C2211yescharacteristic
21C2×D8×C23C2211yescharacteristic
22C2×D8×C23C2211yescharacteristic
23C2×M16×C23C2211yescharacteristic, complemented
24C22×C8×C23C2211yescharacteristic, complemented
25C23×#127—21no—
26C23×#124—21no—
27C23×C22⋊Q8—21no—
28C23×C22⋊Q8C2211yescharacteristic
29C23×C22⋊Q8C2211yescharacteristic, complemented
30C23×#121—21no—
31C23×#121—21no—
32C23×#121—21no—
33C23×#121C2211yescharacteristic
34C23×#121C2211yescharacteristic
35C23×#121C2211yescharacteristic
36C23×#121C2211yescharacteristic
37C23×#121C2211yescharacteristic
38C23×#121C2211yescharacteristic, complemented
39C23×#120—21no—
40C23×#120—21no—
41C23×#117C2211yescharacteristic
42C2×C4⋊C4×C23C2211yescharacteristic, complemented
43C23×#107C2211yescharacteristic, complemented
44C23×#107C2211yescharacteristic, complemented
45C23×#106C2211yescharacteristic, complemented
46C23×#106C2211yescharacteristic, complemented
47C23×#103C2211yescharacteristic, complemented
48C23×#103C2211yescharacteristic, complemented
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic, complemented
54C23×#102C2211yescharacteristic, complemented
55C23×C22⋊C8C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic, complemented
58C23×C22⋊C8C2211yescharacteristic, complemented
59C24×C23—21no—
60#41×C23—41no—
61#41×C23—41no—
62#41×C23—41no—
63#41×C23—41no—
64C2×Q8×C23D411yescharacteristic
65C2×Q8×C23C2311yescharacteristic
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23D411yescharacteristic
77C2×D4×C23D411yescharacteristic
78C2×D4×C23D411yescharacteristic
79C2×D4×C23C2311yescharacteristic
80C2×D4×C23C2311yescharacteristic
81C2×D4×C23C2311yescharacteristic
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23C2311yescharacteristic
85C22×C4×C23C2311yescharacteristic
86C22×C4×C23C2311yescharacteristic
87SD16×C23—41no—
88SD16×C23—41no—
89D8×C23—41no—
90D8×C23—41no—
91M16×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23C2311yescharacteristic
94C2×C8×C23C2311yescharacteristic
95C2×C8×C23C2311yescharacteristic
96C2×C8×C23C2311yescharacteristic
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23C2311yescharacteristic
101C4⋊C4×C23C2311yescharacteristic
102C4⋊C4×C23C2311yescharacteristic
103C4⋊C4×C23C2311yescharacteristic
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23—21no—
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—21no—
110C22⋊C4×C23—21no—
111C22⋊C4×C23—21no—
112C22⋊C4×C23—21no—
113C42×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—42no—
118C23×C23—81no—
119C23×C23—81no—
120C23×C23C2×D411yescharacteristic
121C23×C23C2×D411yescharacteristic
122C23×C23C2×D411yescharacteristic
123Q8×C23—21no—
124Q8×C23—21no—
125Q8×C23—41no—
126D4×C23—21no—
127D4×C23—21no—
128D4×C23—21no—
129D4×C23—21no—
130D4×C23—21no—
131D4×C23—21no—
132D4×C23—41no—
133D4×C23—41no—
134D4×C23—41no—
135D4×C23—42no—
136D4×C23—42no—
137D4×C23—81no—
138D4×C23—81no—
139D4×C23—81no—
140D4×C23—81no—
141D4×C23—81no—
142D4×C23—81no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
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—42no—
155C2×C4×C23—81no—
156C2×C4×C23C2×D411yescharacteristic
157C2×C4×C23C2×D411yescharacteristic
158C2×C4×C23C2×D411yescharacteristic
159C2×C4×C23C2411yescharacteristic
160C8×C23—21no—
161C8×C23—21no—
162C8×C23—21no—
163C8×C23—21no—
164#3198C2311yesmaximal, characteristic, direct product factor, Sylow
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—41no—
175C22×C23—41no—
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—84no—
184C22×C23C22×D411yescharacteristic
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—41no—
192C4×C23—41no—
193C4×C23H(5,𝔽2)11yescharacteristic
194C4×C23H(5,𝔽2)11yescharacteristic
195#537C2×C2311yescharacteristic, complemented
196#535C2×C2311yescharacteristic, complemented
197#485C2×C2311yescharacteristic, complemented
198#482C2×C2311yescharacteristic, complemented
199#481C2×C2311yescharacteristic, complemented
200#480C2×C2311yescharacteristic, complemented
201#469C2×C2311yescharacteristic, complemented
202#468C2×C2311yescharacteristic, complemented
203#466C2×C2311yescharacteristic, complemented
204#465C2×C2311yescharacteristic, complemented
205#450C2×C2311yescharacteristic, complemented
206#449C2×C2311yescharacteristic, complemented
207#449C2×C2311yescharacteristic, complemented
208#447C2×C2311yescharacteristic, complemented
209#408C2×C2311yescharacteristic, complemented
210C2×C23—41no—
211C2×C23—41no—
212C2×C23—41no—
213C2×C23—81no—
214C2×C23—81no—
215C2×C23—81no—
216C2×C23#53411yescharacteristic
217C2×C23#57611yescharacteristic
218C2×C23#57711yescharacteristic
219C2×#41C22×C2311yescharacteristic
220C2×#41C22×C2311yescharacteristic, complemented
221C22×D4C22×C2311yescharacteristic
222C2×SD16C22×C2311yescharacteristic
223C2×SD16C22×C2311yescharacteristic
224C2×D8C22×C2311yescharacteristic
225C2×D8C22×C2311yescharacteristic
226C2×M16C22×C2311yescharacteristic, complemented
227C22×C8C22×C2311yescharacteristic, complemented
228#127—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8C22×C2311yescharacteristic
232C22⋊Q8C22×C2311yescharacteristic, complemented
233#121—21no—
234#121—21no—
235#121—21no—
236#121C22×C2311yescharacteristic
237#121C22×C2311yescharacteristic
238#121C22×C2311yescharacteristic
239#121C22×C2311yescharacteristic
240#121C22×C2311yescharacteristic
241#121C22×C2311yescharacteristic, complemented
242#120—21no—
243#120—21no—
244#117C22×C2311yescharacteristic
245C2×C4⋊C4C22×C2311yescharacteristic, complemented
246#107C22×C2311yescharacteristic, complemented
247#107C22×C2311yescharacteristic, complemented
248#106C22×C2311yescharacteristic, complemented
249#106C22×C2311yescharacteristic, complemented
250#103C22×C2311yescharacteristic, complemented
251#103C22×C2311yescharacteristic, complemented
252#102C22×C2311yescharacteristic
253#102C22×C2311yescharacteristic
254#102C22×C2311yescharacteristic
255#102C22×C2311yescharacteristic
256#102C22×C2311yescharacteristic, complemented
257#102C22×C2311yescharacteristic, complemented
258C22⋊C8C22×C2311yescharacteristic
259C22⋊C8C22×C2311yescharacteristic
260C22⋊C8C22×C2311yescharacteristic, complemented
261C22⋊C8C22×C2311yescharacteristic, complemented
262C23#319811yescharacteristic, direct product factor, Sylow
263C24—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8D4×C2311yescharacteristic
269C2×Q8C23×C2311yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4D4×C2311yescharacteristic
281C2×D4D4×C2311yescharacteristic
282C2×D4D4×C2311yescharacteristic
283C2×D4C23×C2311yescharacteristic
284C2×D4C23×C2311yescharacteristic
285C2×D4C23×C2311yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C23×C2311yescharacteristic
289C22×C4C23×C2311yescharacteristic
290C22×C4C23×C2311yescharacteristic
291SD16—41no—
292SD16—41no—
293D8—41no—
294D8—41no—
295M16—41no—
296C2×C8—41no—
297C2×C8C23×C2311yescharacteristic
298C2×C8C23×C2311yescharacteristic
299C2×C8C23×C2311yescharacteristic
300C2×C8C23×C2311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4C23×C2311yescharacteristic
305C4⋊C4C23×C2311yescharacteristic
306C4⋊C4C23×C2311yescharacteristic
307C4⋊C4C23×C2311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23C2×D4×C2311yescharacteristic
325C23C2×D4×C2311yescharacteristic
326C23C2×D4×C2311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—41no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
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—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4C2×D4×C2311yescharacteristic
361C2×C4C2×D4×C2311yescharacteristic
362C2×C4C2×D4×C2311yescharacteristic
363C2×C4C24×C2311yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—84no—
387C22C22×D4×C2311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4C23×H(5,𝔽2)11yescharacteristic
397C4C23×H(5,𝔽2)11yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—81no—
403C2—81no—
404C2C23×#53411yescharacteristic
405C2C23×#57611yescharacteristic
406C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#537C2semidirect
2. C23×#535C2semidirect
3. C23×#485C2semidirect
4. C23×#482C2semidirect
5. C23×#481C2semidirect
6. C23×#480C2semidirect
7. C23×#469C2semidirect
8. C23×#468C2semidirect
9. C23×#466C2semidirect
10. C23×#465C2semidirect
11. C23×#450C2semidirect
12. C23×#449C2semidirect
13. C23×#449C2semidirect
14. C23×#447C2semidirect
15. C23×#408C2semidirect
17. C2×#41×C23C22semidirect
23. C2×M16×C23C22semidirect
24. C22×C8×C23C22semidirect
29. C23×C22⋊Q8C22semidirect
38. C23×#121C22semidirect
42. C2×C4⋊C4×C23C22semidirect
43. C23×#107C22semidirect
44. C23×#107C22semidirect
45. C23×#106C22semidirect
46. C23×#106C22semidirect
47. C23×#103C22semidirect
48. C23×#103C22semidirect
53. C23×#102C22semidirect
54. C23×#102C22semidirect
57. C23×C22⋊C8C22semidirect
58. C23×C22⋊C8C22semidirect
164. #3198C23direct (N × H)
195. #537C2×C23semidirect
196. #535C2×C23semidirect
197. #485C2×C23semidirect
198. #482C2×C23semidirect
199. #481C2×C23semidirect
200. #480C2×C23semidirect
201. #469C2×C23semidirect
202. #468C2×C23semidirect
203. #466C2×C23semidirect
204. #465C2×C23semidirect
205. #450C2×C23semidirect
206. #449C2×C23semidirect
207. #449C2×C23semidirect
208. #447C2×C23semidirect
209. #408C2×C23semidirect
220. C2×#41C22×C23semidirect
226. C2×M16C22×C23semidirect
227. C22×C8C22×C23semidirect
232. C22⋊Q8C22×C23semidirect
241. #121C22×C23semidirect
245. C2×C4⋊C4C22×C23semidirect
246. #107C22×C23semidirect
247. #107C22×C23semidirect
248. #106C22×C23semidirect
249. #106C22×C23semidirect
250. #103C22×C23semidirect
251. #103C22×C23semidirect
256. #102C22×C23semidirect
257. #102C22×C23semidirect
260. C22⋊C8C22×C23semidirect
261. C22⋊C8C22×C23semidirect
262. C23#3198direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5372
C23×#5352
C23×#4852
C23×#4822
C23×#4812
C23×#4802
C23×#4692
C23×#4682
C23×#4662
C23×#4652
C23×#4502
C23×#4492
C23×#4472
C23×#4082
#319823