Finite Group Databasebeta

← #13075427

Subgroups — #13075427

Order 1408.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#67000C211yesmaximal, characteristic
2C2×#66929C211yesmaximal, characteristic, complemented
3C11×#391C211yesmaximal, characteristic, complemented
4#10622842C211yesmaximal, characteristic, complemented
5#10622607C211yesmaximal, characteristic, complemented
6#10622602C211yesmaximal, characteristic, complemented
7#10622553C211yesmaximal, characteristic, complemented
8#10622289C211yesmaximal, characteristic, complemented
9#10622288C211yesmaximal, characteristic, complemented
10#10622287C211yesmaximal, characteristic, complemented
11#10622285C211yesmaximal, characteristic
12#10622280C211yesmaximal, characteristic, complemented
13#10622278C211yesmaximal, characteristic
14#10622265C211yesmaximal, characteristic, complemented
15#10622260C211yesmaximal, characteristic, complemented
16C22×Q8×C11C2211yescharacteristic
17C22×D44C2211yescharacteristic
18C22×C4×D11C2211yescharacteristic
19C2×C11×C4⋊C4C2211yescharacteristic
20C2×C42×C11C2211yescharacteristic
21#67000—24no—
22C2×(C2×D11)⋊C4C2211yescharacteristic
23C2×(C2×D11)⋊C4C2211yescharacteristic
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×(C2×D11)⋊C4C2211yescharacteristic
31C2×(C2×D11)⋊C4C2211yescharacteristic
32C2×(C2×D11)⋊C4C2211yescharacteristic
33C2×C4⋊Dic11C2211yescharacteristic
34C2×Dic11⋊C4C2211yescharacteristic
35C2×Dic11⋊C4C2211yescharacteristic
36C2×Dic11⋊C4C2211yescharacteristic
37C2×Dic11⋊C4C2211yescharacteristic
38C2×Dic11⋊C4C2211yescharacteristic
39C2×Dic11⋊C4C2211yescharacteristic
40C2×Dic11⋊C4C2211yescharacteristic
41C2×C4×Dic11C2211yescharacteristic
42#66929—24no—
43C11×#95C2211yescharacteristic
44C11×#95C2211yescharacteristic
45C11×#95C2211yescharacteristic
46C11×#95C2211yescharacteristic
47C11⋊#95C2211yescharacteristic
48C11⋊#95C2211yescharacteristic
49C11⋊#95C2211yescharacteristic
50C11⋊#95C2211yescharacteristic
51C11⋊#95C2211yescharacteristic
52C11⋊#95C2211yescharacteristic
53C23×D11C2311yescharacteristic
54C23×D11C2311yescharacteristic
55C2×Q8×C11—22no—
56C2×Q8×C11—22no—
57C2×Q8×C11—41no—
58C22×C4×C11C2311yescharacteristic
59C22×C4×C11C2311yescharacteristic
60C22×C4×C11C2311yescharacteristic
61C22×C4×C11C2311yescharacteristic
62C22×C4×C11C2311yescharacteristic
63C22×C4×C11C2311yescharacteristic
64C22×C4×C11C2311yescharacteristic
65C22×Dic11C2311yescharacteristic
66C22×Dic11C2311yescharacteristic
67C22×Dic11C2311yescharacteristic
68C22×Dic11C2311yescharacteristic
69C22×Dic11C2311yescharacteristic
70C22×Dic11C2311yescharacteristic
71C2×D44—22no—
72C2×D44—22no—
73C2×D44—41no—
74C2×C4×D11—22no—
75C2×C4×D11—41no—
76C2×C4×D11—41no—
77C11×C4⋊C4—41no—
78C42×C11—22no—
79(C2×D11)⋊C4—22no—
80(C2×D11)⋊C4—22no—
81(C2×D11)⋊C4—22no—
82(C2×D11)⋊C4—22no—
83(C2×D11)⋊C4—22no—
84(C2×D11)⋊C4—22no—
85(C2×D11)⋊C4—41no—
86(C2×D11)⋊C4—41no—
87(C2×D11)⋊C4—41no—
88(C2×D11)⋊C4—41no—
89(C2×D11)⋊C4D414yes—
90C4⋊Dic11—41no—
91Dic11⋊C4—22no—
92Dic11⋊C4—22no—
93Dic11⋊C4—22no—
94Dic11⋊C4—22no—
95Dic11⋊C4—22no—
96Dic11⋊C4—41no—
97Dic11⋊C4—41no—
98C4×Dic11—22no—
99#2715—111nomaximal, Sylow
100C23×C11C2411yescharacteristic
101C22×D11—21no—
102C22×D11—21no—
103C22×D11—21no—
104C22×D11—21no—
105C22×D11—21no—
106C22×D11—21no—
107C22×D11—21no—
108C22×D11—21no—
109C22×D11—21no—
110C22×D11—21no—
111C22×D11—21no—
112C22×D11—21no—
113C22×D11—21no—
114C22×D11C2×D412yes—
115Q8×C11—44no—
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×C11—21no—
130C2×C4×C11—21no—
131C2×C4×C11—21no—
132C2×C4×C11—21no—
133C2×C4×C11—21no—
134C2×C4×C11—21no—
135C2×C4×C11C2×D412yes—
136C2×C4×C11#4112yes—
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×Dic11—21no—
149C2×Dic11—21no—
150C2×Dic11—21no—
151C2×Dic11—21no—
152C2×Dic11—21no—
153C2×Dic11C2×D412yes—
154C2×Dic11#4112yes—
155D44—44no—
156C4×D11—44no—
157C2×#126—111no—
158C2×#124—111no—
159#397—111no—
160#397—111no—
161#396—111no—
162#396—111no—
163#395—111no—
164#394—111no—
165#391D1111yescharacteristic, complemented
166#389—111no—
167#388—111no—
168#387—111no—
169#386—111no—
170#385—111no—
171#380—111no—
172C22×C11C22×D411yescharacteristic
173C22×C11C2×#4111yescharacteristic
174C22×C11C2×#4111yescharacteristic
175C22×C11H(5,𝔽2)11yescharacteristic
176C22×C11H(5,𝔽2)11yescharacteristic
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—
186C2×D11—41no—
187C2×D11—41no—
188C2×D11—41no—
189C2×D11—41no—
190C2×D11—41no—
191C2×D11—41no—
192C2×D11—41no—
193C4×C11—22no—
194C4×C11—22no—
195C4×C11—41no—
196C4×C11—41no—
197C4×C11—41no—
198C4×C11—41no—
199C4×C11—41no—
200Dic11—22no—
201Dic11—22no—
202Dic11—41no—
203Dic11—41no—
204Dic11—41no—
205Dic11—41no—
206C22×Q8C2×D1111yescharacteristic
207C22×D4—111no—
208C23×C4—111no—
209#126—224no—
210#124—224no—
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⋊C4—111no—
218C2×C4⋊C4—111no—
219C2×C4⋊C4C2×D1111yescharacteristic
220C2×C22⋊C4—111no—
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×C22⋊C4—111no—
228C2×C22⋊C4—111no—
229C2×C22⋊C4—111no—
230C2×C22⋊C4—111no—
231C2×C42—111no—
232C2×C42C2×D1111yescharacteristic
233#95—111no—
234#95—111no—
235#95—111no—
236#95—111no—
237#95—111no—
238#95—111no—
239#95C2×D1111yescharacteristic
240#95C2×D1111yescharacteristic
241#95C2×D1111yescharacteristic
242#95C2×D1111yescharacteristic
243C2×C11#53711yescharacteristic
244C2×C11#53811yescharacteristic
245C2×C11#54211yescharacteristic
246C2×C11#54611yescharacteristic
247C2×C11#54711yescharacteristic
248C2×C11#55611yescharacteristic
249C2×C11#56311yescharacteristic
250D11—42no—
251D11—81no—
252C24—111no—
253C24—111no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×D4—222no—
258C2×D4—222no—
259C2×D4—441no—
260C22×C4—111no—
261C22×C4—111no—
262C22×C4—111no—
263C22×C4—111no—
264C22×C4—111no—
265C22×C4—111no—
266C22×C4—222no—
267C22×C4—441no—
268C22×C4—441no—
269C22×C4C22×D1111yescharacteristic
270C22×C4C22×D1111yescharacteristic
271C22×C4C22×D1111yescharacteristic
272C22×C4C22×D1111yescharacteristic
273C22×C4C22×D1111yescharacteristic
274C22×C4C22×D1111yescharacteristic
275C22×C4C22×D1111yescharacteristic
276C4⋊C4—41no—
277C4⋊C4—222no—
278C4⋊C4—222no—
279C4⋊C4—222no—
280C4⋊C4—222no—
281C4⋊C4—222no—
282C4⋊C4—441no—
283C4⋊C4—441no—
284C4⋊C4—441no—
285C22⋊C4—114no—
286C22⋊C4—222no—
287C22⋊C4—222no—
288C22⋊C4—222no—
289C22⋊C4—222no—
290C22⋊C4—222no—
291C22⋊C4—222no—
292C22⋊C4—441no—
293C22⋊C4—441no—
294C22⋊C4—441no—
295C22⋊C4—441no—
296C42—22no—
297C42—222no—
298C11#271511yescharacteristic, complemented, Sylow
299C23—112no—
300C23—221no—
301C23—221no—
302C23—221no—
303C23—221no—
304C23—221no—
305C23—221no—
306C23—221no—
307C23—221no—
308C23—221no—
309C23—221no—
310C23—221no—
311C23—221no—
312C23—221no—
313C23C23×D1111yescharacteristic
314Q8—44no—
315D4—444no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—112no—
336C2×C4—112no—
337C2×C4—221no—
338C2×C4—221no—
339C2×C4—221no—
340C2×C4—221no—
341C2×C4—221no—
342C2×C4—221no—
343C2×C4—221no—
344C2×C4—221no—
345C2×C4—221no—
346C2×C4—221no—
347C2×C4—221no—
348C2×C4—221no—
349C2×C4—221no—
350C2×C4—221no—
351C2×C4—221no—
352C2×C4—221no—
353C2×C4—444no—
354C2×C4#433912yes—
355C2×C4D4×D1112yes—
356C22—222no—
357C22—222no—
358C22—222no—
359C22—441no—
360C22—441no—
361C22—441no—
362C22—441no—
363C22—441no—
364C22—441no—
365C22—441no—
366C22—441no—
367C22—441no—
368C22—441no—
369C22—441no—
370C22C2×#433911yescharacteristic
371C22C2×D4×D1111yescharacteristic
372C22#6703711yescharacteristic
373C22#6704011yescharacteristic
374C22#6704011yescharacteristic
375C22#41×D1111yescharacteristic
376C22#6704211yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—222no—
385C4—222no—
386C4—441no—
387C4—441no—
388C4—441no—
389C4—441no—
390C2—442no—
391C2—881no—
392C2#1062302111yescharacteristic
393C2#1062309911yescharacteristic
394C2#1062311311yescharacteristic
395C2#1062316011yescharacteristic
396C2#1062316211yescharacteristic
397C2#1062320911yescharacteristic
398C2#1062326611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC22×D44C22
hypercenter#391D11
hypocenterC11#2715
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#66929C2semidirect
3. C11×#391C2semidirect
4. #10622842C2semidirect
5. #10622607C2semidirect
6. #10622602C2semidirect
7. #10622553C2semidirect
8. #10622289C2semidirect
9. #10622288C2semidirect
10. #10622287C2semidirect
12. #10622280C2semidirect
14. #10622265C2semidirect
15. #10622260C2semidirect
165. #391D11semidirect
298. C11#2715semidirect

Maximal subgroups

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

subgroupindex
C2×#670002
C2×#669292
C11×#3912
#106228422
#106226072
#106226022
#106225532
#106222892
#106222882
#106222872
#106222852
#106222802
#106222782
#106222652
#106222602
#271511