Finite Group Databasebeta

← #13072064

Subgroups — #13072064

Order 1408.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic11C211yesmaximal, characteristic
2C22×Dic11⋊C4C211yesmaximal, characteristic, complemented
3C2×#66931C211yesmaximal, characteristic, complemented
4C2×#66931C211yesmaximal, characteristic, complemented
5C11×#386C211yesmaximal, characteristic
6#10622833C211yesmaximal, characteristic
7#10622825C211yesmaximal, characteristic, complemented
8#10622561C211yesmaximal, characteristic, complemented
9#10622561C211yesmaximal, characteristic, complemented
10#10622555C211yesmaximal, characteristic, complemented
11#10622555C211yesmaximal, characteristic, complemented
12#10622274C211yesmaximal, characteristic, complemented
13#10622270C211yesmaximal, characteristic, complemented
14#10622260C211yesmaximal, characteristic, complemented
15#10622259C211yesmaximal, characteristic, complemented
16C22×D4×C11C2211yescharacteristic
17C23×C4×C11C2211yescharacteristic
18C23×Dic11C2211yescharacteristic
19C23×Dic11C2211yescharacteristic
20C23×Dic11C2211yescharacteristic
21C22×Dic22C2211yescharacteristic
22C2×C11×C22⋊C4C2211yescharacteristic
23C2×C11×C22⋊C4C2211yescharacteristic
24C2×C11×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic11C2211yescharacteristic
26C2×C22⋊Dic11C2211yescharacteristic
27C2×C22⋊Dic11C2211yescharacteristic
28C2×C22⋊Dic11C2211yescharacteristic
29C2×C22⋊Dic11C2211yescharacteristic
30D4×Dic11—24no—
31C2×C4⋊Dic11C2211yescharacteristic
32C2×C4⋊Dic11C2211yescharacteristic
33C2×C4⋊Dic11C2211yescharacteristic
34C2×C4⋊Dic11C2211yescharacteristic
35C2×Dic11⋊C4—24no—
36C2×Dic11⋊C4C2211yescharacteristic
37C2×Dic11⋊C4C2211yescharacteristic
38C2×Dic11⋊C4C2211yescharacteristic
39C2×Dic11⋊C4C2211yescharacteristic
40C2×Dic11⋊C4C2211yescharacteristic
41C2×Dic11⋊C4C2211yescharacteristic
42C2×Dic11⋊C4C2211yescharacteristic
43C2×Dic11⋊C4C2211yescharacteristic
44C2×C4×Dic11C2211yescharacteristic
45#66931—24no—
46#66931—24no—
47C11×#95C2211yescharacteristic
48C11×#95C2211yescharacteristic
49C11⋊#95C2211yescharacteristic
50C11⋊#95C2211yescharacteristic
51C11⋊#95C2211yescharacteristic
52C11⋊#95C2211yescharacteristic
53C11⋊#95C2211yescharacteristic
54C11⋊#95C2211yescharacteristic
55C24×C11C2311yescharacteristic
56C24×C11C2311yescharacteristic
57C2×D4×C11—22no—
58C2×D4×C11—22no—
59C2×D4×C11Q814yescomplemented
60C22×C4×C11—22no—
61C22×C4×C11—22no—
62C22×C4×C11—41no—
63C22×C4×C11C2311yescharacteristic
64C22×C4×C11C2311yescharacteristic
65C22×C4×C11C2311yescharacteristic
66C22×C4×C11C2311yescharacteristic
67C22×C4×C11C2311yescharacteristic
68C22×Dic11—22no—
69C22×Dic11—22no—
70C22×Dic11—22no—
71C22×Dic11—22no—
72C22×Dic11—22no—
73C22×Dic11—22no—
74C22×Dic11—22no—
75C22×Dic11—41no—
76C22×Dic11D414yes—
77C22×Dic11C2311yescharacteristic
78C22×Dic11C2311yescharacteristic
79C22×Dic11C2311yescharacteristic
80C22×Dic11C2311yescharacteristic
81C22×Dic11C2311yescharacteristic
82C22×Dic11C2311yescharacteristic
83C22×Dic11C2311yescharacteristic
84C22×Dic11C2311yescharacteristic
85C2×Dic22—22no—
86C2×Dic22—22no—
87C2×Dic22—22no—
88C11×C22⋊C4—22no—
89C11×C22⋊C4—22no—
90C11×C22⋊C4—41no—
91C22⋊Dic11—22no—
92C22⋊Dic11—22no—
93C22⋊Dic11—22no—
94C22⋊Dic11—22no—
95C22⋊Dic11—41no—
96C4⋊Dic11—22no—
97C4⋊Dic11—22no—
98C4⋊Dic11—22no—
99C4⋊Dic11—41no—
100Dic11⋊C4—22no—
101Dic11⋊C4—22no—
102Dic11⋊C4—22no—
103Dic11⋊C4—22no—
104Dic11⋊C4—22no—
105Dic11⋊C4—22no—
106Dic11⋊C4—41no—
107Dic11⋊C4D414yescomplemented
108C4×Dic11—22no—
109#2417—111nomaximal, Sylow
110C23×C11—21no—
111C23×C11—21no—
112C23×C11—21no—
113C23×C11—21no—
114C23×C11—21no—
115C23×C11—21no—
116C23×C11—21no—
117C23×C11—21no—
118C23×C11—21no—
119C23×C11—21no—
120C23×C11C2×D412yes—
121C23×C11C2×Q812yes—
122C23×C11C2×Q812yes—
123C23×C11#4112yes—
124C23×C11C2411yescharacteristic
125D4×C11—28no—
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×C11—21no—
136C2×C4×C11—21no—
137C2×C4×C11—21no—
138C2×C4×C11—21no—
139C2×C4×C11—44no—
140C2×C4×C11C2×D412yes—
141C2×C4×C11C2×Q812yes—
142C2×Dic11—21no—
143C2×Dic11—21no—
144C2×Dic11—21no—
145C2×Dic11—21no—
146C2×Dic11—28no—
147C2×Dic11—21no—
148C2×Dic11—21no—
149C2×Dic11—21no—
150C2×Dic11—21no—
151C2×Dic11—21no—
152C2×Dic11—21no—
153C2×Dic11—21no—
154C2×Dic11—21no—
155C2×Dic11—21no—
156C2×Dic11—21no—
157C2×Dic11—21no—
158C2×Dic11—21no—
159C2×Dic11—21no—
160C2×Dic11—21no—
161C2×Dic11—21no—
162C2×Dic11—44no—
163C2×Dic11—44no—
164C2×Dic11C2×D412yes—
165C2×Dic11C2×D412yes—
166C2×Dic11C2×D412yes—
167C2×Dic11C2×D412yes—
168C2×Dic11#4112yes—
169Dic22—44no—
170C2×C22⋊Q8—111no—
171C2×C22⋊Q8—111no—
172C2×C4×D4—111no—
173C22×C4⋊C4—111no—
174#398—111no—
175#398—111no—
176#396—111no—
177#396—111no—
178#391—111no—
179#389—111no—
180#386—111no—
181#386D1111yescharacteristic
182#380—111no—
183#380—111no—
184#380—111no—
185C22×C11—22no—
186C22×C11—22no—
187C22×C11—22no—
188C22×C11—22no—
189C22×C11—22no—
190C22×C11—22no—
191C22×C11—22no—
192C22×C11—22no—
193C22×C11—22no—
194C22×C11—41no—
195C22×C11—41no—
196C22×C11—41no—
197C22×C11—41no—
198C22×C11C22⋊Q814yescomplemented
199C22×C11C22×D411yescharacteristic
200C22×C11C22×D411yescharacteristic
201C22×C11C22×Q811yescharacteristic
202C22×C11C2×#4111yescharacteristic
203C22×C11C2×#4111yescharacteristic
204C22×C11H(5,𝔽2)11yescharacteristic
205C22×C112−1+411yescharacteristic
206C4×C11—22no—
207C4×C11—22no—
208C4×C11—41no—
209C4×C11—41no—
210C4×C11—41no—
211Dic11—22no—
212Dic11—22no—
213Dic11—41no—
214Dic11—41no—
215Dic11—41no—
216Dic11—41no—
217Dic11—41no—
218Dic11#12114yes—
219C22×Q8—111no—
220C22×D4C2×D1111yescharacteristic
221C23×C4—111no—
222C23×C4—111no—
223C23×C4—111no—
224C23×C4C2×D1111yescharacteristic
225C22⋊Q8—224no—
226C22⋊Q8—224no—
227C4×D4—224no—
228C2×C4⋊C4—111no—
229C2×C4⋊C4—111no—
230C2×C4⋊C4—111no—
231C2×C4⋊C4—111no—
232C2×C4⋊C4—111no—
233C2×C4⋊C4—111no—
234C2×C4⋊C4—111no—
235C2×C4⋊C4—111no—
236C2×C4⋊C4—111no—
237C2×C4⋊C4—111no—
238C2×C4⋊C4—111no—
239C2×C4⋊C4—111no—
240C2×C4⋊C4—224no—
241C2×C22⋊C4—111no—
242C2×C22⋊C4—111no—
243C2×C22⋊C4—111no—
244C2×C22⋊C4—111no—
245C2×C22⋊C4—111no—
246C2×C22⋊C4C2×D1111yescharacteristic
247C2×C22⋊C4C2×D1111yescharacteristic
248C2×C22⋊C4C2×D1111yescharacteristic
249C2×C42—111no—
250#95—111no—
251#95—111no—
252#95—111no—
253#95—111no—
254#95—111no—
255#95—111no—
256#95C2×D1111yescharacteristic
257#95C2×D1111yescharacteristic
258C2×C11—24no—
259C2×C11—42no—
260C2×C11C2×#12111yescharacteristic
261C2×C11C2×C22⋊Q811yescharacteristic
262C2×C11#53711yescharacteristic
263C2×C11#54611yescharacteristic
264C2×C11D4×Q811yescharacteristic
265C2×C11#55211yescharacteristic
266C2×C11#55411yescharacteristic
267C24C22×D1111yescharacteristic
268C24C22×D1111yescharacteristic
269C2×Q8—222no—
270C2×Q8—222no—
271C2×Q8—222no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4Dic2214yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—111no—
279C22×C4—114no—
280C22×C4—111no—
281C22×C4—111no—
282C22×C4—111no—
283C22×C4—111no—
284C22×C4—111no—
285C22×C4—111no—
286C22×C4—111no—
287C22×C4—222no—
288C22×C4—222no—
289C22×C4—222no—
290C22×C4—222no—
291C22×C4—222no—
292C22×C4—222no—
293C22×C4—222no—
294C22×C4—441no—
295C22×C4C22×D1111yescharacteristic
296C22×C4C22×D1111yescharacteristic
297C22×C4C22×D1111yescharacteristic
298C22×C4C22×D1111yescharacteristic
299C22×C4C22×D1111yescharacteristic
300C4⋊C4—114no—
301C4⋊C4—222no—
302C4⋊C4—222no—
303C4⋊C4—222no—
304C4⋊C4—222no—
305C4⋊C4—222no—
306C4⋊C4—222no—
307C4⋊C4—222no—
308C4⋊C4—222no—
309C4⋊C4—222no—
310C4⋊C4—441no—
311C4⋊C4—441no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—222no—
316C22⋊C4—222no—
317C22⋊C4—222no—
318C22⋊C4—222no—
319C22⋊C4—441no—
320C42—222no—
321C11#241711yescharacteristic, complemented, Sylow
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23C2×Dic2212yes—
333C23C2×Dic2212yes—
334C23D4×D1112yes—
335C23#434112yes—
336C23C23×D1111yescharacteristic
337Q8—444no—
338D4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—112no—
354C2×C4—112no—
355C2×C4—112no—
356C2×C4—112no—
357C2×C4—112no—
358C2×C4—228no—
359C2×C4—221no—
360C2×C4—221no—
361C2×C4—221no—
362C2×C4—221no—
363C2×C4—221no—
364C2×C4—221no—
365C2×C4—221no—
366C2×C4—221no—
367C2×C4—221no—
368C2×C4—221no—
369C2×C4—221no—
370C2×C4—221no—
371C2×C4—221no—
372C2×C4—221no—
373C2×C4—221no—
374C2×C4—221no—
375C2×C4—221no—
376C2×C4—221no—
377C2×C4—221no—
378C2×C4—444no—
379C2×C4—444no—
380C2×C4C2×Dic2212yes—
381C2×C4D4×D1112yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#6693114yescomplemented
396C22C22×Dic2211yescharacteristic
397C22C2×D4×D1111yescharacteristic
398C22C2×D4×D1111yescharacteristic
399C22C2×#434111yescharacteristic
400C22#6703711yescharacteristic
401C22#41×D1111yescharacteristic
402C22#6704311yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—114no—
409C4—222no—
410C4—222no—
411C4—441no—
412C4—441no—
413C4—441no—
414C4—441no—
415C4—441no—
416C2—24no—
417C2—42no—
418C2C2×#6693111yescharacteristic
419C2D4×Dic2211yescharacteristic
420C2#1062307011yescharacteristic
421C2#1062312211yescharacteristic
422C2#1062312811yescharacteristic
423C2D11×#12111yescharacteristic
424C2#1062318111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#386C2
prime radicalC22×D4×C11C22
hypercenter#386D11
hypocenterC11#2417
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic11⋊C4C2semidirect
3. C2×#66931C2semidirect
4. C2×#66931C2semidirect
7. #10622825C2semidirect
8. #10622561C2semidirect
9. #10622561C2semidirect
10. #10622555C2semidirect
11. #10622555C2semidirect
12. #10622274C2semidirect
13. #10622270C2semidirect
14. #10622260C2semidirect
15. #10622259C2semidirect
59. C2×D4×C11Q8semidirect
107. Dic11⋊C4D4semidirect
198. C22×C11C22⋊Q8semidirect
274. C2×D4Dic22semidirect
321. C11#2417semidirect
395. C22#66931semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic112
C22×Dic11⋊C42
C2×#669312
C11×#3862
#106228332
#106228252
#106225612
#106225552
#106222742
#106222702
#106222602
#106222592
#241711