Finite Group Databasebeta

← #13072214

Subgroups — #13072214

Order 1408.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66949C211yesmaximal, characteristic
2C2×#66949C211yesmaximal, characteristic
3C2×#66941C211yesmaximal, characteristic, complemented
4C2×C4×Dic22C211yesmaximal, characteristic, complemented
5C11×#389C211yesmaximal, characteristic, complemented
6#10622607C211yesmaximal, characteristic
7#10622607C211yesmaximal, characteristic
8#10622597C211yesmaximal, characteristic, complemented
9#10622593C211yesmaximal, characteristic, complemented
10#10622585C211yesmaximal, characteristic, complemented
11#10622553C211yesmaximal, characteristic
12#10622282C211yesmaximal, characteristic
13#10622279C211yesmaximal, characteristic
14#10622260C211yesmaximal, characteristic, complemented
15#10622259C211yesmaximal, characteristic, complemented
16C22×C4×D11C2211yescharacteristic
17C22×Dic22C2211yescharacteristic
18C22×Dic22C2211yescharacteristic
19C2×C11×C4⋊C4C2211yescharacteristic
20C2×C11×C4⋊C4C2211yescharacteristic
21C2×C11×C4⋊C4C2211yescharacteristic
22C2×C11×C4⋊C4C2211yescharacteristic
23C2×C42×C11C2211yescharacteristic
24C2×(C2×D11)⋊C4C2211yescharacteristic
25C2×(C2×D11)⋊C4C2211yescharacteristic
26C2×(C2×D11)⋊C4C2211yescharacteristic
27C2×(C2×D11)⋊C4C2211yescharacteristic
28C2×(C2×D11)⋊C4C2211yescharacteristic
29C2×(C2×D11)⋊C4C2211yescharacteristic
30C2×C4⋊Dic11C2211yescharacteristic
31C2×C4⋊Dic11C2211yescharacteristic
32C2×Dic11⋊C4C2211yescharacteristic
33C2×Dic11⋊C4C2211yescharacteristic
34C2×Dic11⋊C4C2211yescharacteristic
35C2×Dic11⋊C4C2211yescharacteristic
36C2×Dic11⋊C4C2211yescharacteristic
37C2×Dic11⋊C4C2211yescharacteristic
38C2×Dic11⋊C4C2211yescharacteristic
39C2×Dic11⋊C4C2211yescharacteristic
40C2×C4×Dic11C2211yescharacteristic
41C2×C4×Dic11C2211yescharacteristic
42C2×C4×Dic11C2211yescharacteristic
43#66949—24no—
44#66949—24no—
45#66941—24no—
46C4×Dic22—24no—
47C11×#95C2211yescharacteristic
48C11×#95C2211yescharacteristic
49C11⋊#95C2211yescharacteristic
50C11⋊#95C2211yescharacteristic
51C11⋊#95C2211yescharacteristic
52C11⋊#95C2211yescharacteristic
53C11⋊#95C2211yescharacteristic
54C11⋊#95C2211yescharacteristic
55C23×D11C2311yescharacteristic
56C22×C4×C11C2311yescharacteristic
57C22×C4×C11C2311yescharacteristic
58C22×C4×C11C2311yescharacteristic
59C22×C4×C11C2311yescharacteristic
60C22×C4×C11C2311yescharacteristic
61C22×C4×C11C2311yescharacteristic
62C22×C4×C11C2311yescharacteristic
63C22×Dic11C2311yescharacteristic
64C22×Dic11C2311yescharacteristic
65C22×Dic11C2311yescharacteristic
66C22×Dic11C2311yescharacteristic
67C22×Dic11C2311yescharacteristic
68C22×Dic11C2311yescharacteristic
69C22×Dic11C2311yescharacteristic
70C2×C4×D11—22no—
71C2×C4×D11—22no—
72C2×C4×D11—22no—
73C2×Dic22—22no—
74C2×Dic22—22no—
75C2×Dic22—22no—
76C2×Dic22—22no—
77C2×Dic22—22no—
78C2×Dic22D414yes—
79C11×C4⋊C4—22no—
80C11×C4⋊C4—22no—
81C11×C4⋊C4—22no—
82C11×C4⋊C4—22no—
83C42×C11—22no—
84(C2×D11)⋊C4—22no—
85(C2×D11)⋊C4—22no—
86(C2×D11)⋊C4—22no—
87(C2×D11)⋊C4—22no—
88(C2×D11)⋊C4—22no—
89(C2×D11)⋊C4Q814yescomplemented
90C4⋊Dic11—22no—
91C4⋊Dic11—22no—
92Dic11⋊C4—22no—
93Dic11⋊C4—22no—
94Dic11⋊C4—22no—
95Dic11⋊C4—22no—
96Dic11⋊C4—22no—
97Dic11⋊C4—22no—
98Dic11⋊C4—22no—
99Dic11⋊C4D414yes—
100C4×Dic11—22no—
101C4×Dic11—22no—
102C4×Dic11—22no—
103#2429—111nomaximal, Sylow
104C23×C11C2411yescharacteristic
105C22×D11—21no—
106C22×D11—21no—
107C22×D11—21no—
108C22×D11—21no—
109C22×D11—21no—
110C22×D11—21no—
111C22×D11C2×Q812yes—
112C2×C4×C11—21no—
113C2×C4×C11—21no—
114C2×C4×C11—21no—
115C2×C4×C11—21no—
116C2×C4×C11—21no—
117C2×C4×C11—21no—
118C2×C4×C11—21no—
119C2×C4×C11—21no—
120C2×C4×C11—21no—
121C2×C4×C11—21no—
122C2×C4×C11—21no—
123C2×C4×C11—21no—
124C2×C4×C11—21no—
125C2×C4×C11—21no—
126C2×C4×C11—21no—
127C2×C4×C11—21no—
128C2×C4×C11—21no—
129C2×C4×C11C2×D412yes—
130C2×C4×C11C2×D412yes—
131C2×C4×C11C2×Q812yes—
132C2×C4×C11#4112yes—
133C2×Dic11—21no—
134C2×Dic11—21no—
135C2×Dic11—21no—
136C2×Dic11—21no—
137C2×Dic11—21no—
138C2×Dic11—21no—
139C2×Dic11—21no—
140C2×Dic11—21no—
141C2×Dic11—21no—
142C2×Dic11—21no—
143C2×Dic11—21no—
144C2×Dic11—21no—
145C2×Dic11—21no—
146C2×Dic11—21no—
147C2×Dic11—21no—
148C2×Dic11C2×D412yes—
149C2×Dic11C2×D412yes—
150C2×Dic11C2×D412yes—
151C2×Dic11C2×D412yes—
152C2×Dic11C2×Q812yes—
153C2×Dic11#4112yes—
154C4×D11—44no—
155Dic22—28no—
156Dic22—44no—
157C2×C4⋊Q8—111no—
158C2×C22⋊Q8—111no—
159C2×C22⋊Q8—111no—
160C2×C4×Q8—111no—
161#398—111no—
162#398—111no—
163#396—111no—
164#396—111no—
165#391—111no—
166#389—111no—
167#389—111no—
168#389D1111yescharacteristic, complemented
169#388—111no—
170#388—111no—
171C4×C22⋊C4—111no—
172C22×C11C22×D411yescharacteristic
173C22×C11C22×D411yescharacteristic
174C22×C11C22×Q811yescharacteristic
175C22×C11C2×#4111yescharacteristic
176C22×C11C2×#4111yescharacteristic
177C22×C112−1+411yescharacteristic
178C22×C112−1+411yescharacteristic
179C2×D11—22no—
180C2×D11—22no—
181C2×D11—22no—
182C2×D11—41no—
183C2×D11—41no—
184C2×D11—41no—
185C2×D11—41no—
186C4×C11—22no—
187C4×C11—22no—
188C4×C11—22no—
189C4×C11—22no—
190C4×C11—41no—
191C4×C11—41no—
192C4×C11—41no—
193Dic11—22no—
194Dic11—22no—
195Dic11—22no—
196Dic11—41no—
197Dic11—41no—
198Dic11—41no—
199Dic11#12114yes—
200C22×Q8—111no—
201C22×Q8—111no—
202C23×C4—111no—
203C4⋊Q8—224no—
204C22⋊Q8—224no—
205C22⋊Q8—224no—
206C4×Q8—224no—
207C2×C4⋊C4—111no—
208C2×C4⋊C4—111no—
209C2×C4⋊C4—111no—
210C2×C4⋊C4—111no—
211C2×C4⋊C4—111no—
212C2×C4⋊C4—111no—
213C2×C4⋊C4—111no—
214C2×C4⋊C4—111no—
215C2×C4⋊C4—111no—
216C2×C4⋊C4—111no—
217C2×C4⋊C4C2×D1111yescharacteristic
218C2×C4⋊C4C2×D1111yescharacteristic
219C2×C4⋊C4C2×D1111yescharacteristic
220C2×C4⋊C4C2×D1111yescharacteristic
221C2×C22⋊C4—111no—
222C2×C22⋊C4—111no—
223C2×C22⋊C4—111no—
224C2×C22⋊C4—111no—
225C2×C22⋊C4—111no—
226C2×C22⋊C4—111no—
227C2×C42—111no—
228C2×C42—111no—
229C2×C42—111no—
230C2×C42C2×D1111yescharacteristic
231#95—111no—
232#95—111no—
233#95—111no—
234#95—111no—
235#95—111no—
236#95—111no—
237#95C2×D1111yescharacteristic
238#95C2×D1111yescharacteristic
239C2×C11C2×#12111yescharacteristic
240C2×C11#53311yescharacteristic
241C2×C11#54111yescharacteristic
242C2×C11#54711yescharacteristic
243C2×C11#54811yescharacteristic
244C2×C11D4×Q811yescharacteristic
245C2×C11D4×Q811yescharacteristic
246D11—42no—
247C24—111no—
248C2×Q8—114no—
249C2×Q8—222no—
250C2×Q8—222no—
251C2×Q8—222no—
252C2×Q8—222no—
253C2×Q8—222no—
254C22×C4—111no—
255C22×C4—111no—
256C22×C4—111no—
257C22×C4—111no—
258C22×C4—111no—
259C22×C4—111no—
260C22×C4—111no—
261C22×C4—222no—
262C22×C4—222no—
263C22×C4—222no—
264C22×C4C22×D1111yescharacteristic
265C22×C4C22×D1111yescharacteristic
266C22×C4C22×D1111yescharacteristic
267C22×C4C22×D1111yescharacteristic
268C22×C4C22×D1111yescharacteristic
269C22×C4C22×D1111yescharacteristic
270C22×C4C22×D1111yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—114no—
276C4⋊C4—222no—
277C4⋊C4—222no—
278C4⋊C4—222no—
279C4⋊C4—222no—
280C4⋊C4—222no—
281C4⋊C4—222no—
282C4⋊C4—222no—
283C4⋊C4—222no—
284C4⋊C4—222no—
285C22⋊C4—114no—
286C22⋊C4—222no—
287C22⋊C4—222no—
288C22⋊C4—222no—
289C22⋊C4—222no—
290C22⋊C4—222no—
291C42—22no—
292C42—222no—
293C42—222no—
294C42—222no—
295C11#242911yescharacteristic, complemented, Sylow
296C23—112no—
297C23—221no—
298C23—221no—
299C23—221no—
300C23—221no—
301C23—221no—
302C23—221no—
303C23C23×D1111yescharacteristic
304Q8—228no—
305Q8—444no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—112no—
324C2×C4—112no—
325C2×C4—112no—
326C2×C4—112no—
327C2×C4—112no—
328C2×C4—112no—
329C2×C4—221no—
330C2×C4—221no—
331C2×C4—221no—
332C2×C4—221no—
333C2×C4—221no—
334C2×C4—221no—
335C2×C4—221no—
336C2×C4—221no—
337C2×C4—221no—
338C2×C4—221no—
339C2×C4—221no—
340C2×C4—221no—
341C2×C4—221no—
342C2×C4—221no—
343C2×C4—221no—
344C2×C4—444no—
345C2×C4#433912yes—
346C2×C4D4×D1112yes—
347C2×C4D4×D1112yes—
348C2×C4Q8×D1112yes—
349C22—222no—
350C22—222no—
351C22—222no—
352C22—441no—
353C22—441no—
354C22—441no—
355C22—441no—
356C22C2×#433911yescharacteristic
357C22C2×D4×D1111yescharacteristic
358C22C2×D4×D1111yescharacteristic
359C22C2×Q8×D1111yescharacteristic
360C22#6704011yescharacteristic
361C22#41×D1111yescharacteristic
362C22#6704311yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—114no—
371C4—222no—
372C4—222no—
373C4—222no—
374C4—441no—
375C4—441no—
376C4—441no—
377C2—442no—
378C2#1062303811yescharacteristic
379C2#1062308311yescharacteristic
380C2#1062310911yescharacteristic
381C2D11×#12111yescharacteristic
382C2#1062316311yescharacteristic
383C2#1062322711yescharacteristic
384C2#1062325811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#389C2
prime radicalC23×D11C23
hypercenter#389D11
hypocenterC11#2429
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#66941C2semidirect
4. C2×C4×Dic22C2semidirect
5. C11×#389C2semidirect
8. #10622597C2semidirect
9. #10622593C2semidirect
10. #10622585C2semidirect
14. #10622260C2semidirect
15. #10622259C2semidirect
89. (C2×D11)⋊C4Q8semidirect
168. #389D11semidirect
295. C11#2429semidirect

Maximal subgroups

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

subgroupindex
C2×#669492
C2×#669412
C2×C4×Dic222
C11×#3892
#106226072
#106225972
#106225932
#106225852
#106225532
#106222822
#106222792
#106222602
#106222592
#242911