Finite Group Databasebeta

← C23×#8505 #19156131

Subgroups — C23×#8505 #19156131

Order 5888.

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

normal subgroups only

This group has 648 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×#3531C211yesmaximal, characteristic
2C23×#1995C211yesmaximal, characteristic, complemented
3C23×#1544C211yesmaximal, characteristic, complemented
4C23×#1500C212yesmaximal, complemented
5C23×#1495C212yesmaximal, complemented
6C23×#564C411yescharacteristic, complemented
7C23×#563—22no—
8C23×#563—22no—
9C23×#563C412yescomplemented
10C23×#562—22no—
11C23×#562C411yescharacteristic, complemented
12C23×#533—22no—
13C23×#533C2211yescharacteristic
14C23×#529—22no—
15C23×#529—22no—
16C23×#529C2212yes—
17C23×#431C2212yes—
18C23×#423C2211yescharacteristic
19C23×#376C2211yescharacteristic
20C23×#339—22no—
21C23×#332—22no—
22C23×#328—22no—
23C23×#327—22no—
24C23×Q8⋊C8—22no—
25C23×D4⋊C8—22no—
26C2×M16×C23—21no—
27C23×C4⋊Q8—22no—
28C23×C4⋊Q8—22no—
29C23×C4⋊Q8D412yes—
30C23×#126—22no—
31C23×#126—22no—
32C23×#126—22no—
33C23×#126—22no—
34C23×#126—22no—
35C23×#125—22no—
36C23×#125—22no—
37C23×#125—22no—
38C23×#125—22no—
39C23×#125C2×C411yescharacteristic
40C23×#125C2×C411yescharacteristic
41C23×#125C2×C412yes—
42C23×#124—22no—
43C23×#124—22no—
44C23×#124C2×C412yes—
45C23×#123—22no—
46C23×#123—22no—
47C23×#123—22no—
48C23×#123—22no—
49C23×#123—22no—
50C23×C22⋊Q8—22no—
51C23×C22⋊Q8—22no—
52C23×C22⋊Q8—22no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×C22⋊Q8D412yes—
58C23×#121—22no—
59C23×#121—22no—
60C23×#121D412yes—
61C4×Q8×C23—22no—
62C4×Q8×C23—22no—
63C4×Q8×C23D412yes—
64C4×D4×C23—22no—
65C4×D4×C23—22no—
66C4×D4×C23D412yes—
67C23×#117—22no—
68C23×#117—22no—
69C23×#117D412yes—
70C2×C42×C23—22no—
71C2×C42×C23—21no—
72C2×C42×C23C2311yescharacteristic
73C23×C4⋊C8—22no—
74C23×C4⋊C8—22no—
75C23×C22⋊C8—22no—
76C4×C8×C23—22no—
77C23×#95—22no—
78C2×Q8×C23—22no—
79C2×Q8×C23—22no—
80C2×Q8×C23C22⋊C412yes—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23C22⋊C412yes—
84C22×C4×C23—22no—
85C22×C4×C23—22no—
86C22×C4×C23—22no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D412yes—
90M16×C23—42no—
91C2×C8×C23—42no—
92C2×C8×C23—41no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—22no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—22no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23C22⋊C412yes—
107C4⋊C4×C23C22⋊C412yes—
108C4⋊C4×C23C22⋊C412yes—
109C4⋊C4×C23C22⋊C412yes—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—22no—
113C22⋊C4×C23—22no—
114C22⋊C4×C23—22no—
115C22⋊C4×C23—22no—
116C22⋊C4×C23—22no—
117C22⋊C4×C23—22no—
118C22⋊C4×C23—22no—
119C22⋊C4×C23—22no—
120C42×C23—22no—
121C42×C23—22no—
122C42×C23—22no—
123C42×C23—22no—
124C42×C23—22no—
125C42×C23—41no—
126C42×C23C22×C411yescharacteristic
127C42×C23C2×D411yescharacteristic
128C42×C23C2×D412yes—
129#8505C2311yesmaximal, characteristic, direct product factor, Sylow
130C23×C23—22no—
131C23×C23#12011yescharacteristic
132Q8×C23—42no—
133Q8×C23—82no—
134Q8×C23—82no—
135D4×C23—42no—
136D4×C23—82no—
137D4×C23—82no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—22no—
143C2×C4×C23—22no—
144C2×C4×C23—22no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—82no—
150C2×C4×C23—82no—
151C2×C4×C23—82no—
152C2×C4×C23C2×C22⋊C411yescharacteristic
153C2×C4×C23C2×C22⋊C412yes—
154C2×C4×C23#12012yes—
155C2×C4×C23#12011yescharacteristic
156C8×C23—82no—
157C8×C23—81no—
158#3531C2×C2311yescharacteristic
159#1995C2×C2311yescharacteristic, complemented
160#1544C2×C2311yescharacteristic, complemented
161#1500C2×C2312yescomplemented
162#1495C2×C2312yescomplemented
163C22×C23—21no—
164C22×C23—22no—
165C22×C23—42no—
166C22×C23—42no—
167C22×C23—42no—
168C22×C23#37911yescharacteristic
169C4×C23—21no—
170C4×C23—22no—
171C4×C23—22no—
172C4×C23—42no—
173C4×C23—42no—
174C4×C23—42no—
175C4×C23—42no—
176C4×C23—42no—
177C4×C23—82no—
178C4×C23#45411yescharacteristic
179C4×C23#45611yescharacteristic
180#564C4×C2311yescharacteristic, complemented
181#563—22no—
182#563—22no—
183#563C4×C2312yescomplemented
184#562—22no—
185#562C4×C2311yescharacteristic, complemented
186#533—22no—
187#533C22×C2311yescharacteristic
188#529—22no—
189#529—22no—
190#529C22×C2312yes—
191#431C22×C2312yes—
192#423C22×C2311yescharacteristic
193#376C22×C2311yescharacteristic
194#339—22no—
195#332—22no—
196#328—22no—
197#327—22no—
198Q8⋊C8—22no—
199D4⋊C8—22no—
200C2×C23—41no—
201C2×C23—82no—
202C2×C23#179311yescharacteristic
203C2×C23#179412yes—
204C2×M16—21no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8D4×C2312yes—
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#125—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125C2×C4×C2312yes—
218#125C2×C4×C2311yescharacteristic
219#125C2×C4×C2311yescharacteristic
220#124—22no—
221#124—22no—
222#124C2×C4×C2312yes—
223#123—22no—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8D4×C2312yes—
236#121—22no—
237#121—22no—
238#121D4×C2312yes—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8D4×C2312yes—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4D4×C2312yes—
245#117—22no—
246#117—22no—
247#117D4×C2312yes—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42C23×C2311yescharacteristic
251C4⋊C8—22no—
252C4⋊C8—22no—
253C22⋊C8—22no—
254C4×C8—22no—
255#95—22no—
256C23#850511yescharacteristic, direct product factor, Sylow
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8C22⋊C4×C2312yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊C4×C2312yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4C2×D4×C2312yes—
269M16—42no—
270C2×C8—42no—
271C2×C8—41no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4C22⋊C4×C2312yes—
286C4⋊C4C22⋊C4×C2312yes—
287C4⋊C4C22⋊C4×C2312yes—
288C4⋊C4C22⋊C4×C2312yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42C22×C4×C2311yescharacteristic
306C42C2×D4×C2312yes—
307C42C2×D4×C2311yescharacteristic
308C23—22no—
309C23C23×#12011yescharacteristic
310Q8—42no—
311Q8—82no—
312Q8—82no—
313D4—42no—
314D4—82no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C2×C22⋊C4×C2311yescharacteristic
331C2×C4C2×C22⋊C4×C2312yes—
332C2×C4C23×#12012yes—
333C2×C4C23×#12011yescharacteristic
334C8—82no—
335C8—81no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22C23×#37911yescharacteristic
342C4—21no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4C23×#45411yescharacteristic
352C4C23×#45611yescharacteristic
353C2—41no—
354C2—82no—
355C2C23×#179311yescharacteristic
356C2C23×#179412yes—
subgroupG/H
centerC2×C4×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#379
prime radicalC23×#3531C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1995C2semidirect
3. C23×#1544C2semidirect
4. C23×#1500C2semidirect
5. C23×#1495C2semidirect
6. C23×#564C4semidirect
9. C23×#563C4semidirect
11. C23×#562C4semidirect
129. #8505C23direct (N × H)
159. #1995C2×C23semidirect
160. #1544C2×C23semidirect
161. #1500C2×C23semidirect
162. #1495C2×C23semidirect
180. #564C4×C23semidirect
183. #563C4×C23semidirect
185. #562C4×C23semidirect
256. C23#8505direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35312
C23×#19952
C23×#15442
C23×#15002
C23×#14952
#850523