Finite Group Databasebeta

← #11920211

Subgroups — #11920211

Order 1152.

600 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 68 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×C4×(C22⋊C9)⋊C2C211yesmaximal, characteristic
2C2×(C22⋊C9)⋊Q8C211yesmaximal, characteristic, complemented
3C4⋊C4×C22⋊C9C211yesmaximal, characteristic, complemented
4(C2×(C22⋊C9)⋊C2)⋊C4C212yesmaximal
5C4⋊((C22⋊C9)⋊C4)C212yesmaximal, complemented
6#82194—31nomaximal
7C22×(C22⋊C9)⋊C2C2211yescharacteristic
8C2×C4×C22⋊C9C2212yes—
9C2×C4×C22⋊C9C2211yescharacteristic
10C2×(C22⋊C9)⋊C4C2212yes—
11C2×(C22⋊C9)⋊C4C2211yescharacteristic
12C4×(C22⋊C9)⋊C2—22no—
13(C22⋊C9)⋊Q8—22no—
14#63459—41nomaximal
15C22×C3×C4⋊C4D311yescharacteristic
16C2×C4×#67—31no—
17C2×#924—31no—
18C2×#897—31no—
19#5213—32no—
20#5212—32no—
21#4994—32no—
22#4987—31no—
23#4984—31no—
24#4981—32no—
25#4973—31no—
26C22×C22⋊C9C2311yescharacteristic
27C2×(C22⋊C9)⋊C2—21no—
28C2×(C22⋊C9)⋊C2—21no—
29C2×(C22⋊C9)⋊C2Q812yescomplemented
30C2×C4×D9—41no—
31C2×Dic18—41no—
32C4×C22⋊C9—22no—
33C4×C22⋊C9D412yes—
34(C22⋊C9)⋊C4—22no—
35(C22⋊C9)⋊C4—21no—
36C9×C4⋊C4—41no—
37(C2×D9)⋊C4—42no—
38C4⋊Dic9—42no—
39#2417—91noSylow
40C23×C3×C4C2×D312yes—
41C23×C3×C4C2×D311yescharacteristic
42C22×#67—31no—
43C22×C4×D3—31no—
44C22×Dic6—31no—
45C2×C3×C4⋊C4—31no—
46C2×C3×C4⋊C4—31no—
47C2×C3×C4⋊C4—32no—
48C2×#217—31no—
49C2×#217—32no—
50C4×#67—62no—
51C4×#67—62no—
52C2×(C2×D3)⋊C4—32no—
53C2×(C2×D3)⋊C4—31no—
54C2×(C2×D3)⋊C4—32no—
55C2×C4⋊Dic3—32no—
56C2×C4⋊Dic3—31no—
57C2×C4⋊Dic3—32no—
58#924—62no—
59#924—62no—
60C2×Dic3⋊C4—31no—
61C2×Dic3⋊C4—32no—
62C2×C4×Dic3—31no—
63#897—61no—
64#897—62no—
65#897—61no—
66C3⋊#95—32no—
67C3⋊#95—32no—
68C3⋊#95—32no—
69C3⋊#95—32no—
70C22×D9—41no—
71C2×C22⋊C9C2×D411yescharacteristic
72C2×C22⋊C9C2×Q811yescharacteristic
73C2×C22⋊C9#4111yescharacteristic
74(C22⋊C9)⋊C2—22no—
75C2×C4×C9—41no—
76C2×C4×C9—42no—
77C2×Dic9—41no—
78C2×Dic9—42no—
79C4×D9—82no—
80Dic18—82no—
81C2×C22⋊Q8—91no—
82C2×C22⋊Q8—91no—
83C2×C4×D4—91no—
84C22×C4⋊C4D911yescharacteristic, complemented
85#398—92no—
86#396—92no—
87#391—91no—
88#389—91no—
89#386—92no—
90#380—91no—
91#380—92no—
92C24×C3C22×D311yescharacteristic
93C23×D3—31no—
94C22×C3×C4—22no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—32no—
98C22×C3×C4—32no—
99C22×C3×C4—32no—
100C22×C3×C4—62no—
101C22×C3×C4D1212yes—
102C2×#67—32no—
103C2×#67—32no—
104C2×#67—61no—
105C2×#67—61no—
106C2×#67—61no—
107C2×#67—61no—
108C22×Dic3—31no—
109C22×Dic3—31no—
110C22×Dic3—32no—
111C22×Dic3—31no—
112C22×Dic3—32no—
113C2×C4×D3—61no—
114C2×C4×D3—62no—
115C2×C4×D3—62no—
116C2×C4×D3—61no—
117C2×Dic6—61no—
118C2×Dic6—62no—
119C2×Dic6—62no—
120C2×Dic6—61no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4S411yescharacteristic
125#217—62no—
126#217—61no—
127#217—62no—
128#217—61no—
129(C2×D3)⋊C4—62no—
130(C2×D3)⋊C4—62no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—122no—
133C4⋊Dic3—62no—
134C4⋊Dic3—62no—
135C4⋊Dic3—62no—
136C4⋊Dic3—122no—
137Dic3⋊C4—64no—
138Dic3⋊C4—62no—
139C4×Dic3—62no—
140C22×C9—41no—
141C2×D9—42no—
142C2×D9—81no—
143C2×D9—81no—
144C22⋊C9C22⋊Q811yescharacteristic, complemented
145C4×C9—42no—
146C4×C9—82no—
147Dic9—82no—
148Dic9—81no—
149C22×Q8—91no—
150C22×D4—91no—
151C23×C4—91no—
152C23×C4C2×D912yes—
153C23×C4C2×D911yescharacteristic
154C22⋊Q8—181no—
155C22⋊Q8—182no—
156C22⋊Q8—182no—
157C22⋊Q8—182no—
158C22⋊Q8—181no—
159C4×D4—182no—
160C4×D4—182no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—32no—
164C2×C4⋊C4—91no—
165C2×C4⋊C4—91no—
166C2×C4⋊C4—92no—
167C2×C4⋊C4—92no—
168C2×C4⋊C4—92no—
169C2×C22⋊C4—91no—
170C2×C22⋊C4—92no—
171C2×C22⋊C4—91no—
172C2×C22⋊C4—92no—
173C2×C22⋊C4—92no—
174C2×C42—91no—
175#95—92no—
176#95—92no—
177#95—92no—
178#95—92no—
179C23×C3—31no—
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—31no—
183C23×C3C2×D1211yescharacteristic
184C23×C3#23711yescharacteristic
185C23×C3D3×Q811yescharacteristic
186C22×D3—32no—
187C22×D3—61no—
188C22×D3—61no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—61no—
192C22×D3—61no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4C2×S412yes—
204C2×C3×C4C2×S411yescharacteristic
205#67—62no—
206#67—62no—
207#67—62no—
208#67—62no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—62no—
224C4×D3—122no—
225C4×D3—122no—
226Dic6—122no—
227Dic6—122no—
228C2×C9—41no—
229C2×C9—41no—
230C2×C9—41no—
231D9—82no—
232C24—91no—
233C24C22×D911yescharacteristic
234C2×Q8—181no—
235C2×Q8—182no—
236C2×Q8—182no—
237C2×Q8—181no—
238C2×D4—92no—
239C2×D4—92no—
240C2×D4—181no—
241C2×D4—181no—
242C2×D4—181no—
243C2×D4—181no—
244C22×C4—22no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—62no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—92no—
254C22×C4—92no—
255C22×C4—91no—
256C22×C4—182no—
257C22×C4—181no—
258C22×C4—182no—
259C22×C4—181no—
260C22×C4D3612yes—
261C4⋊C4—32no—
262C4⋊C4—31no—
263C4⋊C4—61no—
264C4⋊C4—182no—
265C4⋊C4—184no—
266C4⋊C4—182no—
267C4⋊C4—182no—
268C4⋊C4—182no—
269C4⋊C4—362no—
270C4⋊C4(C22⋊C9)⋊C211yescharacteristic, complemented
271C22⋊C4—182no—
272C22⋊C4—182no—
273C22⋊C4—181no—
274C22⋊C4—182no—
275C22⋊C4—181no—
276C22⋊C4—182no—
277C22⋊C4—182no—
278C22⋊C4—362no—
279C42—182no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—61no—
290C22×C3#89711yescharacteristic
291C22×C3C22×S411yescharacteristic
292C2×D3—62no—
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—121no—
299C3×C4—22no—
300C3×C4—32no—
301C3×C4—62no—
302C3×C4C4⋊S412yes—
303Dic3—62no—
304Dic3—121no—
305Dic3—122no—
306Dic3—122no—
307Dic3—121no—
308C9—41noSylow
309C23—31no—
310C23—31no—
311C23—31no—
312C23—31no—
313C23—92no—
314C23—181no—
315C23—181no—
316C23—181no—
317C23—181no—
318C23—181no—
319C23—181no—
320C23C2×D3611yescharacteristic
321C23#369911yescharacteristic
322C23Q8×D911yescharacteristic
323Q8—362no—
324Q8—362no—
325D4—182no—
326D4—182no—
327D4—182no—
328D4—182no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—92no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—182no—
344C2×C4—181no—
345C2×C4—182no—
346C2×C4—182no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—182no—
350C2×C4—181no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—181no—
354C2×C4—362no—
355C2×C4—362no—
356C2×C4C2×(C22⋊C9)⋊C212yes—
357C2×C4C2×(C22⋊C9)⋊C211yescharacteristic
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3C2×C4⋊S411yescharacteristic
363C2×C3#591411yescharacteristic
364C2×C3Q8×S411yescharacteristic
365D3—122no—
366C22—31no—
367C22—31no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—31no—
372C22—31no—
373C22—31no—
374C22—31no—
375C22—61no—
376C22—182no—
377C22—182no—
378C22—182no—
379C22—361no—
380C22—361no—
381C22—361no—
382C22—361no—
383C22#6345911yescharacteristic, complemented
384C22C22×(C22⋊C9)⋊C211yescharacteristic
385C4—22no—
386C4—32no—
387C4—62no—
388C4—182no—
389C4—361no—
390C4—362no—
391C4—361no—
392C4—362no—
393C4C4⋊((C22⋊C9)⋊C2)12yes—
394C3#8531211yescharacteristic
395C2—31no—
396C2—31no—
397C2—31no—
398C2—31no—
399C2—362no—
400C2C2×C4⋊((C22⋊C9)⋊C2)11yescharacteristic
401C2#1059308911yescharacteristic
402C2Q8×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC22C22×(C22⋊C9)⋊C2
commutator, cosocleC22×C22⋊C9C23
FrattiniC22×C3C22×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×(C22⋊C9)⋊C2C22
hypercenterC4⋊C4(C22⋊C9)⋊C2
hypocenterC22⋊C9C22⋊Q8
LCSG ▷ C22×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊C9)⋊Q8C2semidirect
3. C4⋊C4×C22⋊C9C2semidirect
5. C4⋊((C22⋊C9)⋊C4)C2semidirect
29. C2×(C22⋊C9)⋊C2Q8semidirect
84. C22×C4⋊C4D9semidirect
144. C22⋊C9C22⋊Q8semidirect
270. C4⋊C4(C22⋊C9)⋊C2semidirect
383. C22#63459semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C22⋊C9)⋊C22
C2×(C22⋊C9)⋊Q82
C4⋊C4×C22⋊C92
(C2×(C22⋊C9)⋊C2)⋊C42
C4⋊((C22⋊C9)⋊C4)2
#821943
#634594