Finite Group Databasebeta

← #124×C22⋊C9 #11919701

Subgroups — #124×C22⋊C9 #11919701

Order 1152.

804 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C2×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2C2×D4×C22⋊C9C211yesmaximal, characteristic
3C22⋊C4×C22⋊C9C214yesmaximal, complemented
4C42×C22⋊C9C211yesmaximal, characteristic, complemented
5C22×C3×#124C311yesmaximal, characteristic
6C23×C22⋊C9C2212yes—
7Q8×C22⋊C9—22no—
8D4×C22⋊C9—22no—
9C2×C4×C22⋊C9C2212yes—
10C2×C4×C22⋊C9C2212yes—
11C2×C4×C22⋊C9C2211yescharacteristic
12C9×#124—41nomaximal
13C23×C3×Q8C2×C311yescharacteristic
14C23×C3×D4C2×C311yescharacteristic
15C2×C3×#124—31no—
16C2×C3×#124—31no—
17C2×C3×#124—34no—
18C2×C3×#124—31no—
19C2×C3×#124—31no—
20C22×C3×C22⋊C4C2×C314yes—
21C22×C3×C42C2×C311yescharacteristic
22C22×C22⋊C9—22no—
23C22×C22⋊C9—24no—
24C22×C22⋊C9C2311yescharacteristic
25C2×Q8×C9—41no—
26C2×D4×C9—41no—
27C4×C22⋊C9—22no—
28C4×C22⋊C9—22no—
29C4×C22⋊C9D412yes—
30C9×C22⋊C4—44no—
31C42×C9—41no—
32C22×#124C911yescharacteristic, complemented, Sylow
33C25×C3C22×C312yes—
34C22×C3×Q8—22no—
35C22×C3×Q8—31no—
36C22×C3×Q8—32no—
37C22×C3×Q8—31no—
38C22×C3×Q8—62no—
39C22×C3×D4—22no—
40C22×C3×D4—31no—
41C22×C3×D4—31no—
42C22×C3×D4—32no—
43C22×C3×D4—62no—
44C23×C3×C4C22×C312yes—
45C23×C3×C4C22×C312yes—
46C23×C3×C4C22×C311yescharacteristic
47C3×#124—34no—
48C3×#124—34no—
49C3×#124—32no—
50C3×#124—34no—
51C3×#124—31no—
52C3×#124—31no—
53C3×#124—34no—
54C3×#124—31no—
55C3×#124A411yescharacteristic
56C2×C3×C22⋊C4—34no—
57C2×C3×C22⋊C4—34no—
58C2×C3×C22⋊C4—34no—
59C2×C3×C22⋊C4—34no—
60C2×C3×C42—31no—
61C2×C3×C42—31no—
62C2×C3×C42—32no—
63C23×C9—42no—
64C2×C22⋊C9—42no—
65C2×C22⋊C9C2×D411yescharacteristic
66C2×C22⋊C9#4112yes—
67Q8×C9—82no—
68D4×C9—82no—
69C2×C4×C9—41no—
70C2×C4×C9—42no—
71C2×C4×C9—42no—
72C23×Q8C2×C911yescharacteristic, complemented
73C23×D4C2×C911yescharacteristic
74C2×#124—31no—
75C2×#124—31no—
76C2×#124—34no—
77C2×#124—31no—
78C2×#124—31no—
79C22×C22⋊C4C2×C914yescomplemented
80C22×C42C2×C911yescharacteristic, complemented
81C24×C3—22no—
82C24×C3—24no—
83C24×C3—32no—
84C24×C3—32no—
85C24×C3—64no—
86C24×C3—62no—
87C24×C3C23×C311yescharacteristic
88C2×C3×Q8—32no—
89C2×C3×Q8—32no—
90C2×C3×Q8—31no—
91C2×C3×Q8—62no—
92C2×C3×Q8—62no—
93C2×C3×Q8—62no—
94C2×C3×Q8—64no—
95C2×C3×Q8—62no—
96C2×C3×Q8—64no—
97C2×C3×Q8C2×A411yescharacteristic
98C2×C3×D4—32no—
99C2×C3×D4—32no—
100C2×C3×D4—31no—
101C2×C3×D4—62no—
102C2×C3×D4—62no—
103C2×C3×D4—64no—
104C2×C3×D4—62no—
105C2×C3×D4—64no—
106C2×C3×D4—62no—
107C2×C3×D4C2×A411yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—62no—
118C22×C3×C4—62no—
119C22×C3×C4C3×D412yes—
120C3×C22⋊C4—38no—
121C3×C22⋊C4—34no—
122C3×C22⋊C4—34no—
123C3×C22⋊C4—34no—
124C3×C22⋊C4C2×A414yes—
125C3×C42—32no—
126C3×C42—32no—
127C3×C42—31no—
128C3×C42C2×A411yescharacteristic
129C22×C9—41no—
130C22×C9—82no—
131C22×C9—84no—
132C22⋊C9#12411yescharacteristic, direct product factor
133C4×C9—42no—
134C4×C9—82no—
135C4×C9—82no—
136C25C22×C912yes—
137C22×Q8—22no—
138C22×Q8—31no—
139C22×Q8—32no—
140C22×Q8—31no—
141C22×Q8—62no—
142C22×D4—22no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—32no—
146C22×D4—62no—
147C23×C4C22×C912yes—
148C23×C4C22×C912yes—
149C23×C4C22×C911yescharacteristic
150#124—34no—
151#124—34no—
152#124—32no—
153#124—34no—
154#124—31no—
155#124—31no—
156#124—34no—
157#124—31no—
158#124C22⋊C911yescharacteristic, direct product factor
159C2×C22⋊C4—34no—
160C2×C22⋊C4—34no—
161C2×C22⋊C4—34no—
162C2×C22⋊C4—34no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—32no—
166C23×C3—31no—
167C23×C3—32no—
168C23×C3—31no—
169C23×C3—32no—
170C23×C3—42no—
171C23×C3—62no—
172C23×C3—62no—
173C23×C3—64no—
174C23×C3—64no—
175C23×C3—62no—
176C23×C3—62no—
177C23×C3—64no—
178C23×C3—64no—
179C23×C3—122no—
180C23×C3—124no—
181C23×C3—124no—
182C23×C3C2×C3×D411yescharacteristic
183C23×C3C3×#4112yes—
184C23×C3C22×A412yes—
185C3×Q8—22no—
186C3×Q8—64no—
187C3×Q8—64no—
188C3×Q8—62no—
189C3×D4—22no—
190C3×D4—64no—
191C3×D4—64no—
192C3×D4—62no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—62no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4C22×A412yes—
209C2×C3×C4C22×A412yes—
210C2×C3×C4C22×A411yescharacteristic
211C2×C9—41no—
212C2×C9—42no—
213C2×C9—162no—
214C24—22no—
215C24—24no—
216C24—32no—
217C24—32no—
218C24—62no—
219C24—64no—
220C24C23×C911yescharacteristic
221C2×Q8—32no—
222C2×Q8—32no—
223C2×Q8—31no—
224C2×Q8—62no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×Q8—64no—
228C2×Q8—62no—
229C2×Q8—64no—
230C2×Q8C2×C22⋊C911yescharacteristic, complemented
231C2×D4—32no—
232C2×D4—32no—
233C2×D4—31no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—64no—
237C2×D4—62no—
238C2×D4—64no—
239C2×D4—62no—
240C2×D4C2×C22⋊C911yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4D4×C912yes—
253C22⋊C4—38no—
254C22⋊C4—34no—
255C22⋊C4—34no—
256C22⋊C4—34no—
257C22⋊C4C2×C22⋊C914yescomplemented
258C42—32no—
259C42—32no—
260C42—31no—
261C42C2×C22⋊C911yescharacteristic, complemented
262C22×C3—22no—
263C22×C3—24no—
264C22×C3—31no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—32no—
270C22×C3—32no—
271C22×C3—62no—
272C22×C3—64no—
273C22×C3—122no—
274C22×C3—122no—
275C22×C3—124no—
276C22×C3—122no—
277C22×C3—122no—
278C22×C3—124no—
279C22×C3C3×#12411yescharacteristic
280C22×C3C23×A411yescharacteristic
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—32no—
284C3×C4—62no—
285C3×C4—62no—
286C3×C4D4×A412yes—
287C9—41noSylow
288C23—31no—
289C23—32no—
290C23—31no—
291C23—32no—
292C23—42no—
293C23—62no—
294C23—62no—
295C23—64no—
296C23—64no—
297C23—62no—
298C23—62no—
299C23—64no—
300C23—64no—
301C23—122no—
302C23—124no—
303C23—124no—
304C23C2×D4×C911yescharacteristic
305C23C9×#4112yes—
306C23C22×C22⋊C912yes—
307Q8—22no—
308Q8—64no—
309Q8—64no—
310Q8—62no—
311D4—22no—
312D4—64no—
313D4—64no—
314D4—62no—
315C2×C4—32no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4C22×C22⋊C912yes—
331C2×C4C22×C22⋊C912yes—
332C2×C4C22×C22⋊C911yescharacteristic
333C2×C3—31no—
334C2×C3—32no—
335C2×C3—31no—
336C2×C3—42no—
337C2×C3—122no—
338C2×C3C2×D4×A411yescharacteristic
339C2×C3A4×#4112yes—
340C22—22no—
341C22—24no—
342C22—31no—
343C22—32no—
344C22—32no—
345C22—31no—
346C22—31no—
347C22—32no—
348C22—32no—
349C22—62no—
350C22—64no—
351C22—122no—
352C22—122no—
353C22—124no—
354C22—122no—
355C22—122no—
356C22—124no—
357C22C9×#12411yescharacteristic, complemented
358C22C23×C22⋊C911yescharacteristic
359C4—22no—
360C4—22no—
361C4—32no—
362C4—62no—
363C4—62no—
364C4D4×C22⋊C912yes—
365C3A4×#12411yescharacteristic
366C2—31no—
367C2—31no—
368C2—32no—
369C2—42no—
370C2—122no—
371C2C2×D4×C22⋊C911yescharacteristic
372C2#41×C22⋊C912yes—
subgroupG/H
center, FrattiniC22×C3C23×A4
commutatorC24C23×C9
FittingC22×C3×#124C3
socle, cosocleC24×C3C23×C3
prime radicalC23×C3×D4C2×C3
hypercenterC3×#124A4
hypocenterC22C9×#124
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C22⋊C9C2semidirect
3. C22⋊C4×C22⋊C9C2semidirect
4. C42×C22⋊C9C2semidirect
32. C22×#124C9semidirect
72. C23×Q8C2×C9semidirect
79. C22×C22⋊C4C2×C9semidirect
80. C22×C42C2×C9semidirect
132. C22⋊C9#124direct (N × H)
158. #124C22⋊C9direct (N × H)
230. C2×Q8C2×C22⋊C9semidirect
257. C22⋊C4C2×C22⋊C9semidirect
261. C42C2×C22⋊C9semidirect
357. C22C9×#124semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C92
C2×D4×C22⋊C92
C22⋊C4×C22⋊C92
C42×C22⋊C92
C22×C3×#1243
C9×#1244