Finite Group Databasebeta

← C23×#12724 #19160350

Subgroups — C23×#12724 #19160350

Order 5888.

670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3509C211yesmaximal, characteristic
2C23×#2135C211yesmaximal, characteristic
3C23×#2129C211yesmaximal, characteristic
4C23×#2123C212yesmaximal
5C23×#1827C212yesmaximal
6C23×#557—22no—
7C23×#556—22no—
8C23×#552—24no—
9C23×#541—22no—
10C23×#536C2211yescharacteristic
11C23×#536C2211yescharacteristic
12C23×#533—22no—
13C23×#533C411yescharacteristic, complemented
14C2×C23×C4⋊Q8C411yescharacteristic, complemented
15C23×#519C411yescharacteristic, complemented
16C23×#519C411yescharacteristic, complemented
17C4×#41×C23C2211yescharacteristic
18C23×#413C2212yes—
19C23×#410C2212yes—
20C23×#356—22no—
21C23×#354—22no—
22C23×#352—24no—
23C23×#339—22no—
24C23×H(3,ℤ/4)—22no—
25C2×#41×C23C2311yescharacteristic
26C22×Q8×C23C2×C411yescharacteristic
27C22×Q8×C23C2×C411yescharacteristic
28C2×M16×C23—22no—
29C23×C4⋊Q8—22no—
30C23×C4⋊Q8—21no—
31C23×C4⋊Q8—21no—
32C23×C4⋊Q8—21no—
33C23×C4⋊Q8—21no—
34C23×C4⋊Q8D412yes—
35C23×#126—24no—
36C23×#126—24no—
37C23×#125—24no—
38C23×#125—24no—
39C23×#125—24no—
40C23×#125—21no—
41C23×#124—21no—
42C23×#124D412yes—
43C23×#123—22no—
44C23×#123D414yes—
45C23×C22⋊Q8—24no—
46C23×C22⋊Q8—22no—
47C23×C22⋊Q8—22no—
48C23×C22⋊Q8—22no—
49C4×Q8×C23—24no—
50C4×Q8×C23—22no—
51C4×Q8×C23—21no—
52C4×Q8×C23—22no—
53C4×Q8×C23—22no—
54C4×Q8×C23—22no—
55C4×Q8×C23—22no—
56C4×D4×C23—22no—
57C4×D4×C23—21no—
58C23×#117—24no—
59C23×#117—22no—
60C23×#117—22no—
61C23×#117—22no—
62C23×#117C2×C411yescharacteristic
63C23×#117C2×C411yescharacteristic
64C23×#117C2×C411yescharacteristic
65C23×#117D412yes—
66C2×C4⋊C4×C23—22no—
67C2×C42×C23C2×C411yescharacteristic
68C2×C42×C23D412yes—
69C23×#101—22no—
70C23×#100—22no—
71C23×#99—22no—
72C23×#99—22no—
73#41×C23—42no—
74C2×Q8×C23—22no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23—21no—
78C2×Q8×C23—42no—
79C2×Q8×C23—42no—
80C2×Q8×C23C22⋊C412yes—
81C2×Q8×C23C2×D411yescharacteristic
82C2×D4×C23C2×D411yescharacteristic
83C2×D4×C23C2×D412yes—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—22no—
87C22×C4×C23C22⋊C412yes—
88C22×C4×C23C22×C411yescharacteristic
89C22×C4×C23C2×D412yes—
90M16×C23—42no—
91M16×C23—42no—
92C2×C8×C23—42no—
93C4⋊C4×C23—24no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—24no—
97C4⋊C4×C23—24no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—24no—
105C4⋊C4×C23—24no—
106C4⋊C4×C23—42no—
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—24no—
110C22⋊C4×C23—24no—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23C22⋊C412yes—
114C42×C23—24no—
115C42×C23—24no—
116C42×C23—21no—
117C42×C23—21no—
118C42×C23—22no—
119C42×C23—42no—
120C42×C23C22⋊C412yes—
121C42×C23C22⋊C411yescharacteristic
122C42×C23C22⋊C411yescharacteristic
123C42×C23C22⋊C412yes—
124#12724C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23C2×C22⋊C411yescharacteristic
126C23×C23#12012yes—
127Q8×C23—41no—
128Q8×C23—42no—
129Q8×C23—44no—
130Q8×C23—42no—
131Q8×C23—42no—
132Q8×C23—42no—
133D4×C23—42no—
134D4×C23—41no—
135C2×C4×C23—21no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—22no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—22no—
143C2×C4×C23—22no—
144C2×C4×C23—21no—
145C2×C4×C23—42no—
146C2×C4×C23—44no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—82no—
150C2×C4×C23C2×C22⋊C411yescharacteristic
151C2×C4×C23C2×C22⋊C411yescharacteristic
152C2×C4×C23#12012yes—
153C8×C23—82no—
154#3509C2×C2311yescharacteristic
155#2135C2×C2311yescharacteristic
156#2129C2×C2311yescharacteristic
157#2123C2×C2312yes—
158#1827C2×C2312yes—
159C22×C23—21no—
160C22×C23—22no—
161C22×C23—41no—
162C22×C23—42no—
163C22×C23#37911yescharacteristic
164C4×C23—22no—
165C4×C23—22no—
166C4×C23—21no—
167C4×C23—42no—
168C4×C23—41no—
169C4×C23—42no—
170C4×C23—42no—
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—162no—
174C4×C23#41012yes—
175#557—22no—
176#556—22no—
177#552—24no—
178#541—22no—
179#536C22×C2311yescharacteristic
180#536C22×C2311yescharacteristic
181#533—22no—
182#533C4×C2311yescharacteristic, complemented
183C2×C4⋊Q8C4×C2311yescharacteristic, complemented
184#519C4×C2311yescharacteristic, complemented
185#519C4×C2311yescharacteristic, complemented
186C4×#41C22×C2311yescharacteristic
187#413C22×C2312yes—
188#410C22×C2312yes—
189#356—22no—
190#354—22no—
191#352—24no—
192#339—22no—
193H(3,ℤ/4)—22no—
194C2×C23—21no—
195C2×C23—42no—
196C2×C23—41no—
197C2×C23#178211yescharacteristic
198C2×#41C23×C2311yescharacteristic
199C22×Q8C2×C4×C2311yescharacteristic
200C22×Q8C2×C4×C2311yescharacteristic
201C2×M16—22no—
202C4⋊Q8—22no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8D4×C2312yes—
208#126—24no—
209#126—24no—
210#125—24no—
211#125—24no—
212#125—24no—
213#125—21no—
214#124—21no—
215#124D4×C2312yes—
216#123—22no—
217#123D4×C2314yes—
218C22⋊Q8—24no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C4×Q8—24no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—21no—
231#117—24no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117C2×C4×C2311yescharacteristic
236#117C2×C4×C2311yescharacteristic
237#117C2×C4×C2311yescharacteristic
238#117D4×C2312yes—
239C2×C4⋊C4—22no—
240C2×C42C2×C4×C2311yescharacteristic
241C2×C42D4×C2312yes—
242#101—22no—
243#100—22no—
244#99—22no—
245#99—22no—
246C23#1272411yescharacteristic, direct product factor, Sylow
247#41—42no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8C22⋊C4×C2312yes—
255C2×Q8C2×D4×C2311yescharacteristic
256C2×D4C2×D4×C2311yescharacteristic
257C2×D4C2×D4×C2312yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4C22⋊C4×C2312yes—
262C22×C4C22×C4×C2311yescharacteristic
263C22×C4C2×D4×C2312yes—
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4C22⋊C4×C2312yes—
288C42—24no—
289C42—24no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—42no—
294C42C22⋊C4×C2311yescharacteristic
295C42C22⋊C4×C2311yescharacteristic
296C42C22⋊C4×C2312yes—
297C42C22⋊C4×C2312yes—
298C23C2×C22⋊C4×C2311yescharacteristic
299C23C23×#12012yes—
300Q8—41no—
301Q8—42no—
302Q8—42no—
303Q8—44no—
304Q8—42no—
305Q8—42no—
306D4—42no—
307D4—41no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4C2×C22⋊C4×C2311yescharacteristic
324C2×C4C2×C22⋊C4×C2311yescharacteristic
325C2×C4C23×#12012yes—
326C8—82no—
327C22—21no—
328C22—22no—
329C22—41no—
330C22—42no—
331C22C23×#37911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—162no—
342C4C23×#41012yes—
343C2—21no—
344C2—42no—
345C2—41no—
346C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC2×#41×C23C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C4×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
13. C23×#533C4semidirect
14. C2×C23×C4⋊Q8C4semidirect
15. C23×#519C4semidirect
16. C23×#519C4semidirect
124. #12724C23direct (N × H)
182. #533C4×C23semidirect
183. C2×C4⋊Q8C4×C23semidirect
184. #519C4×C23semidirect
185. #519C4×C23semidirect
246. C23#12724direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35092
C23×#21352
C23×#21292
C23×#21232
C23×#18272
#1272423