Finite Group Databasebeta

← C23×#8497 #19156123

Subgroups — C23×#8497 #19156123

Order 5888.

820 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3543C211yesmaximal, characteristic
2C23×#1961C211yesmaximal, characteristic, complemented
3C23×#1555C211yesmaximal, characteristic, complemented
4C23×#1532C212yesmaximal, complemented
5C23×#1512C212yesmaximal, complemented
6C23×#560—22no—
7C23×#559C411yescharacteristic, complemented
8C23×#550C412yescomplemented
9C23×#546—22no—
10D42×C23—22no—
11D42×C23C411yescharacteristic, complemented
12C23×#540—22no—
13C23×#535—22no—
14C23×#534—22no—
15C2×C23×#127C2211yescharacteristic
16C23×#529C2212yes—
17C23×#432C2212yescomplemented
18C2×C23×#102—22no—
19C2×C23×#97C2211yescharacteristic, complemented
20C23×#382C2211yescharacteristic, complemented
21C23×#331—22no—
22C23×#330—22no—
23C23×#329—22no—
24C23×#328—22no—
25C23×#327—22no—
26C2×#41×C23—22no—
27C22×D4×C23—22no—
28C22×D4×C23—22no—
29C22×D4×C23—22no—
30C22×D4×C23D412yes—
31C22×C8×C23—21no—
32C23×#127—22no—
33C23×#127—22no—
34C23×#127C2×C411yescharacteristic
35C23×#127C2×C411yescharacteristic
36C23×#127D412yes—
37C23×#126—22no—
38C23×#125D412yes—
39C23×#124—22no—
40C23×#124D412yes—
41C23×#123—24no—
42C23×#123—22no—
43C23×#123—22no—
44C23×#123—22no—
45C23×C22⋊Q8D412yes—
46C23×#121—24no—
47C23×#121—24no—
48C23×#121—24no—
49C23×#121—24no—
50C23×#121—22no—
51C23×#121—22no—
52C23×#121D412yes—
53C23×#120—24no—
54C23×#120—22no—
55C23×#120—24no—
56C23×#120—22no—
57C4×Q8×C23C2×C412yes—
58C4×D4×C23—22no—
59C4×D4×C23—22no—
60C4×D4×C23—22no—
61C4×D4×C23C2×C412yes—
62C23×#117—22no—
63C2×C4⋊C4×C23—21no—
64C2×C22⋊C4×C23—22no—
65C2×C42×C23C2311yescharacteristic
66C23×C4⋊C8—22no—
67C23×#102—44no—
68C23×C22⋊C8—22no—
69C23×#97—22no—
70C23×#97—22no—
71C23×#95—22no—
72C24×C23—22no—
73C24×C23—22no—
74#41×C23—84no—
75C2×Q8×C23C22⋊C412yes—
76C2×D4×C23—22no—
77C2×D4×C23—24no—
78C2×D4×C23—24no—
79C2×D4×C23—24no—
80C2×D4×C23—22no—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23—24no—
84C2×D4×C23—42no—
85C2×D4×C23—42no—
86C2×D4×C23—84no—
87C2×D4×C23—84no—
88C2×D4×C23—82no—
89C2×D4×C23—82no—
90C2×D4×C23C22⋊C412yes—
91C2×D4×C23C22⋊C412yes—
92C2×D4×C23C22⋊C412yes—
93C22×C4×C23—24no—
94C22×C4×C23—22no—
95C22×C4×C23—22no—
96C22×C4×C23—41no—
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23C2×D412yes—
99C2×C8×C23—22no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C2×C8×C23—42no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—22no—
107C4⋊C4×C23—42no—
108C4⋊C4×C23C22⋊C412yes—
109C4⋊C4×C23C22⋊C412yes—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—24no—
113C22⋊C4×C23—24no—
114C22⋊C4×C23—22no—
115C22⋊C4×C23—22no—
116C22⋊C4×C23—22no—
117C22⋊C4×C23—22no—
118C22⋊C4×C23—22no—
119C42×C23—22no—
120C42×C23C22×C411yescharacteristic
121C42×C23C2×D411yescharacteristic
122C42×C23C2×D412yes—
123#8497C2311yesmaximal, characteristic, direct product factor, Sylow
124C23×C23—22no—
125C23×C23—22no—
126C23×C23—22no—
127C23×C23—42no—
128C23×C23—42no—
129C23×C23—84no—
130C23×C23—82no—
131C23×C23—84no—
132C23×C23#12011yescharacteristic
133Q8×C23—42no—
134D4×C23—42no—
135D4×C23—42no—
136D4×C23—42no—
137D4×C23—44no—
138D4×C23—44no—
139D4×C23—42no—
140D4×C23—42no—
141D4×C23—82no—
142D4×C23—84no—
143D4×C23—84no—
144D4×C23—84no—
145D4×C23—84no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—42no—
152C2×C4×C23—44no—
153C2×C4×C23—82no—
154C2×C4×C23—84no—
155C2×C4×C23—82no—
156C2×C4×C23—81no—
157C2×C4×C23C2×C22⋊C412yes—
158C2×C4×C23C2×C22⋊C411yescharacteristic
159C2×C4×C23#12011yescharacteristic
160C2×C4×C23#12012yes—
161C8×C23—42no—
162C8×C23—82no—
163#3543C2×C2311yescharacteristic
164#1961C2×C2311yescharacteristic, complemented
165#1555C2×C2311yescharacteristic, complemented
166#1532C2×C2312yescomplemented
167#1512C2×C2312yescomplemented
168C22×C23—22no—
169C22×C23—42no—
170C22×C23—42no—
171C22×C23—42no—
172C22×C23—42no—
173C22×C23—44no—
174C22×C23—44no—
175C22×C23—42no—
176C22×C23—162no—
177C22×C23—162no—
178C22×C23—164no—
179C22×C23#37911yescharacteristic
180C22×C23#45311yescharacteristic
181C22×C23#45511yescharacteristic
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—22no—
185C4×C23—22no—
186C4×C23—42no—
187C4×C23—42no—
188C4×C23—42no—
189C4×C23—161no—
190#560—22no—
191#559C4×C2311yescharacteristic, complemented
192#550C4×C2312yescomplemented
193#546—22no—
194D42—22no—
195D42C4×C2311yescharacteristic, complemented
196#540—22no—
197#535—22no—
198#534—22no—
199C2×#127C22×C2311yescharacteristic
200#529C22×C2312yes—
201#432C22×C2312yescomplemented
202C2×#102—22no—
203C2×#97C22×C2311yescharacteristic, complemented
204#382C22×C2311yescharacteristic, complemented
205#331—22no—
206#330—22no—
207#329—22no—
208#328—22no—
209#327—22no—
210C2×C23—21no—
211C2×C23—21no—
212C2×C23—82no—
213C2×C23—82no—
214C2×C23—82no—
215C2×C23#179211yescharacteristic
216C2×C23#179512yes—
217C2×#41—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4D4×C2312yes—
222C22×C8—21no—
223#127—22no—
224#127—22no—
225#127C2×C4×C2311yescharacteristic
226#127C2×C4×C2311yescharacteristic
227#127D4×C2312yes—
228#126—22no—
229#125D4×C2312yes—
230#124—22no—
231#124D4×C2312yes—
232#123—22no—
233#123—24no—
234#123—22no—
235#123—22no—
236C22⋊Q8D4×C2312yes—
237#121—24no—
238#121—24no—
239#121—24no—
240#121—24no—
241#121—22no—
242#121—22no—
243#121D4×C2312yes—
244#120—22no—
245#120—24no—
246#120—24no—
247#120—22no—
248C4×Q8C2×C4×C2312yes—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4C2×C4×C2312yes—
253#117—22no—
254C2×C4⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C42C23×C2311yescharacteristic
257C4⋊C8—22no—
258#102—44no—
259C22⋊C8—22no—
260#97—22no—
261#97—22no—
262#95—22no—
263C23#849711yescharacteristic, direct product factor, Sylow
264C24—22no—
265C24—22no—
266#41—84no—
267C2×Q8C22⋊C4×C2312yes—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C22⋊C4×C2312yes—
283C2×D4C22⋊C4×C2312yes—
284C2×D4C22⋊C4×C2312yes—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4C2×D4×C2311yescharacteristic
290C22×C4C2×D4×C2312yes—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4C22⋊C4×C2312yes—
301C4⋊C4C22⋊C4×C2312yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42—22no—
312C42C22×C4×C2311yescharacteristic
313C42C2×D4×C2312yes—
314C42C2×D4×C2311yescharacteristic
315C23—22no—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—84no—
321C23—82no—
322C23—84no—
323C23C23×#12011yescharacteristic
324Q8—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—44no—
329D4—44no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4C2×C22⋊C4×C2312yes—
349C2×C4C2×C22⋊C4×C2311yescharacteristic
350C2×C4C23×#12011yescharacteristic
351C2×C4C23×#12012yes—
352C8—42no—
353C8—82no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—162no—
363C22—162no—
364C22—164no—
365C22C23×#37911yescharacteristic
366C22C23×#45311yescharacteristic
367C22C23×#45511yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—161no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—82no—
380C2—82no—
381C2C23×#179211yescharacteristic
382C2C23×#179512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3543C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1961C2semidirect
3. C23×#1555C2semidirect
4. C23×#1532C2semidirect
5. C23×#1512C2semidirect
7. C23×#559C4semidirect
8. C23×#550C4semidirect
11. D42×C23C4semidirect
17. C23×#432C22semidirect
19. C2×C23×#97C22semidirect
20. C23×#382C22semidirect
123. #8497C23direct (N × H)
164. #1961C2×C23semidirect
165. #1555C2×C23semidirect
166. #1532C2×C23semidirect
167. #1512C2×C23semidirect
191. #559C4×C23semidirect
192. #550C4×C23semidirect
195. D42C4×C23semidirect
201. #432C22×C23semidirect
203. C2×#97C22×C23semidirect
204. #382C22×C23semidirect
263. C23#8497direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35432
C23×#19612
C23×#15552
C23×#15322
C23×#15122
#849723