Finite Group Databasebeta

← #44408100

Subgroups — #44408100

Order 29056.

516 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300283C211yesmaximal, characteristic, complemented
2C2×#20300281C211yesmaximal, characteristic, complemented
3#28408932C211yesmaximal, characteristic
4#28408932C211yesmaximal, characteristic
5#28408932C211yesmaximal, characteristic
6#28408929C211yesmaximal, characteristic, complemented
7#28408929C211yesmaximal, characteristic, complemented
8#28408927C211yesmaximal, characteristic, complemented
9#28408925C211yesmaximal, characteristic, complemented
10#28408924C211yesmaximal, characteristic, complemented
11#28408923C211yesmaximal, characteristic, complemented
12#28408921C211yesmaximal, characteristic, complemented
13#28408920C211yesmaximal, characteristic, complemented
14C42×Dic227C211yesmaximal, characteristic, complemented
15#388×C227C211yesmaximal, characteristic, complemented
16C22×C22⋊D227C2211yescharacteristic
17#20300283—24no—
18#20300281—24no—
19C2×C4⋊Dic227C2211yescharacteristic
20C2×C4⋊Dic227C2211yescharacteristic
21C2×Dic227⋊C4C2211yescharacteristic
22C2×Dic227⋊C4C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×Dic227⋊C4C2211yescharacteristic
27C2×C22⋊Dic227C2211yescharacteristic
28C2×C22⋊Dic227C2211yescharacteristic
29C2×C22⋊Dic227C2211yescharacteristic
30C2×(C2×D227)⋊C4C2211yescharacteristic
31C2×(C2×D227)⋊C4C2211yescharacteristic
32C2×(C2×D227)⋊C4C2211yescharacteristic
33C2×(C2×D227)⋊C4C2211yescharacteristic
34C2×(C2×D227)⋊C4C2211yescharacteristic
35C2×(C2×D227)⋊C4C2211yescharacteristic
36C2×C4×Dic227C2211yescharacteristic
37C2×C4×Dic227C2211yescharacteristic
38C2×C4×Dic227C2211yescharacteristic
39C2×C4×Dic227C2211yescharacteristic
40C2×C4×Dic227C2211yescharacteristic
41C2×C4×Dic227C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C227⋊#95C2211yescharacteristic
46C2×C4⋊C4×C227C2211yescharacteristic
47C2×C22⋊C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50C2×C42×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53C23×D227C2311yescharacteristic
54C2×C22⋊D227—22no—
55C2×C22⋊D227—22no—
56C2×C22⋊D227—22no—
57C22×Dic227C2311yescharacteristic
58C22×Dic227C2311yescharacteristic
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C4⋊Dic227—22no—
65C4⋊Dic227—22no—
66Dic227⋊C4—22no—
67Dic227⋊C4—22no—
68Dic227⋊C4—22no—
69Dic227⋊C4—22no—
70Dic227⋊C4—22no—
71Dic227⋊C4—22no—
72C22⋊Dic227—22no—
73C22⋊Dic227—22no—
74C22⋊Dic227—22no—
75(C2×D227)⋊C4—22no—
76(C2×D227)⋊C4—22no—
77(C2×D227)⋊C4—22no—
78(C2×D227)⋊C4—22no—
79(C2×D227)⋊C4—22no—
80(C2×D227)⋊C4—22no—
81C4×Dic227—22no—
82C4×Dic227—22no—
83C4×Dic227—22no—
84C4×Dic227—22no—
85C4×Dic227—22no—
86C4×Dic227D414yescomplemented
87C24×C227C2311yescharacteristic
88C22×C4×C227C2311yescharacteristic
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91C22×C4×C227C2311yescharacteristic
92C22×C4×C227C2311yescharacteristic
93C22×C4×C227C2311yescharacteristic
94C4⋊C4×C227—22no—
95C22⋊C4×C227—22no—
96C22⋊C4×C227—22no—
97C22⋊C4×C227—22no—
98C42×C227—22no—
99C22×D227—21no—
100C22×D227—21no—
101C22×D227—21no—
102C22×D227—21no—
103C22×D227—21no—
104C22×D227—21no—
105C22×D227—21no—
106C22⋊D227—44no—
107C2×Dic227—21no—
108C2×Dic227—21no—
109C2×Dic227—21no—
110C2×Dic227—21no—
111C2×Dic227—21no—
112C2×Dic227—21no—
113C2×Dic227—21no—
114C2×Dic227—21no—
115C2×Dic227—21no—
116C2×Dic227—21no—
117C2×Dic227—21no—
118C2×Dic227—21no—
119C2×Dic227—21no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227C2×D412yes—
123C2×Dic227C2×D412yes—
124C2×Dic227#4112yes—
125C2×Dic227#4112yes—
126C2×Dic227#4112yes—
127C2×Dic227#4112yes—
128C23×C227—21no—
129C23×C227—21no—
130C23×C227—21no—
131C23×C227—21no—
132C23×C227—21no—
133C23×C227—21no—
134C23×C227—21no—
135C23×C227C2411yescharacteristic
136C2×C4×C227—21no—
137C2×C4×C227—21no—
138C2×C4×C227—21no—
139C2×C4×C227—21no—
140C2×C4×C227—21no—
141C2×C4×C227—21no—
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—21no—
145C2×C4×C227—21no—
146C2×C4×C227—21no—
147C2×C4×C227—21no—
148C2×C4×C227—21no—
149C2×C4×C227—21no—
150C2×C4×C227—21no—
151C2×C4×C227C2×D412yes—
152C2×C4×C227#4112yes—
153C2×C4×C227#4112yes—
154C22×C227—41no—
155C22×C227—41no—
156C22×C227—41no—
157C22×C227—41no—
158C22×C227—41no—
159C22×C227—41no—
160C22×C227—41no—
161C22×C227C22×D411yescharacteristic
162C22×C227C2×#4111yescharacteristic
163C22×C227C2×#4111yescharacteristic
164C22×C227C2×#4111yescharacteristic
165C22×C227C2×#4111yescharacteristic
166C22×C227C2×#4111yescharacteristic
167C22×C227C2×#4111yescharacteristic
168C2×D227—41no—
169C2×D227—41no—
170C2×D227—41no—
171C2×D227—41no—
172C2×D227—41no—
173C2×D227—41no—
174C2×D227—41no—
175C4×C227—22no—
176C4×C227—22no—
177C4×C227—22no—
178C4×C227—41no—
179C4×C227—41no—
180C4×C227—41no—
181Dic227—22no—
182Dic227—22no—
183Dic227—22no—
184Dic227—41no—
185Dic227—41no—
186Dic227—41no—
187Dic227#12614yescomplemented
188C2×C227—81no—
189C2×C227C2×#12611yescharacteristic
190C2×C227#52911yescharacteristic
191C2×C227#52911yescharacteristic
192C2×C227#52911yescharacteristic
193C2×C227#53211yescharacteristic
194C2×C227#53211yescharacteristic
195C2×C227#53211yescharacteristic
196D227—81no—
197C227#286011yescharacteristic, complemented, Sylow
198#2860—2271nomaximal, Sylow
199C2×#126—2271no—
200C2×#126—2271no—
201#390—2271no—
202#390—2271no—
203#390—2271no—
204#389—2271no—
205#389—2271no—
206#389—2271no—
207#388—2271no—
208#388—2271no—
209#388—2271no—
210#388—2271no—
211#388—2271no—
212#388D22711yescharacteristic, complemented
213C43—2271no—
214C22×D4—2271no—
215#126—4544no—
216#126—4544no—
217C2×C4⋊C4—2271no—
218C2×C4⋊C4—2271no—
219C2×C4⋊C4—2271no—
220C2×C4⋊C4—2271no—
221C2×C4⋊C4—2271no—
222C2×C4⋊C4—2271no—
223C2×C4⋊C4—2271no—
224C2×C4⋊C4—2271no—
225C2×C4⋊C4C2×D22711yescharacteristic
226C2×C22⋊C4—2271no—
227C2×C22⋊C4—2271no—
228C2×C22⋊C4—2271no—
229C2×C22⋊C4—2271no—
230C2×C22⋊C4—2271no—
231C2×C22⋊C4—2271no—
232C2×C22⋊C4—2271no—
233C2×C22⋊C4—2271no—
234C2×C22⋊C4—2271no—
235C2×C22⋊C4C2×D22711yescharacteristic
236C2×C22⋊C4C2×D22711yescharacteristic
237C2×C22⋊C4C2×D22711yescharacteristic
238C2×C42—2271no—
239C2×C42—2271no—
240C2×C42—2271no—
241C2×C42—2271no—
242C2×C42—2271no—
243C2×C42—2271no—
244C2×C42C2×D22711yescharacteristic
245#95—2271no—
246#95—2271no—
247#95—2271no—
248#95—2271no—
249#95C2×D22711yescharacteristic
250#95C2×D22711yescharacteristic
251C24—2271no—
252C24C22×D22711yescharacteristic
253C2×D4—4542no—
254C2×D4—4542no—
255C2×D4—4542no—
256C22×C4—2271no—
257C22×C4—2271no—
258C22×C4—2271no—
259C22×C4—2271no—
260C22×C4—2271no—
261C22×C4—2271no—
262C22×C4—2271no—
263C22×C4C22×D22711yescharacteristic
264C22×C4C22×D22711yescharacteristic
265C22×C4C22×D22711yescharacteristic
266C22×C4C22×D22711yescharacteristic
267C22×C4C22×D22711yescharacteristic
268C22×C4C22×D22711yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—4542no—
271C4⋊C4—4542no—
272C4⋊C4—4542no—
273C4⋊C4—4542no—
274C4⋊C4—4542no—
275C4⋊C4—4542no—
276C4⋊C4—4542no—
277C4⋊C4—4542no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—4542no—
282C22⋊C4—4542no—
283C22⋊C4—4542no—
284C22⋊C4—4542no—
285C22⋊C4—4542no—
286C22⋊C4—4542no—
287C22⋊C4—4542no—
288C22⋊C4—4542no—
289C22⋊C4—4542no—
290C42—22no—
291C42—4542no—
292C42—4542no—
293C42—4542no—
294C42—4542no—
295C42—4542no—
296C42—2274no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—4541no—
305C23—4541no—
306C23—4541no—
307C23—4541no—
308C23—4541no—
309C23—4541no—
310C23—4541no—
311C23C23×D22711yescharacteristic
312D4—9084no—
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—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—4541no—
329C2×C4—4541no—
330C2×C4—4541no—
331C2×C4—4541no—
332C2×C4—4541no—
333C2×C4—4541no—
334C2×C4—4541no—
335C2×C4—4541no—
336C2×C4—4541no—
337C2×C4—4541no—
338C2×C4—4541no—
339C2×C4—4541no—
340C2×C4—4541no—
341C2×C4—4541no—
342C2×C4—4541no—
343C2×C4—2272no—
344C2×C4—2272no—
345C2×C4—2272no—
346C2×C4—2272no—
347C2×C4—2272no—
348C2×C4—2272no—
349C2×C4D4×D22712yes—
350C2×C4#1681388312yes—
351C2×C4#1681388412yes—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—9081no—
360C22—9081no—
361C22—9081no—
362C22—9081no—
363C22—9081no—
364C22—9081no—
365C22—9081no—
366C22C2×D4×D22711yescharacteristic
367C22#41×D22711yescharacteristic
368C22#41×D22711yescharacteristic
369C22#41×D22711yescharacteristic
370C22#41×D22711yescharacteristic
371C22C2×#1681388311yescharacteristic
372C22C2×#1681388411yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—9081no—
380C4—9081no—
381C4—9081no—
382C4—4542no—
383C4—4542no—
384C4—4542no—
385C4—2274no—
386C2—81no—
387C2—18161no—
388C2#126×D22711yescharacteristic
389C2#2840955911yescharacteristic
390C2#2840956011yescharacteristic
391C2#2840956211yescharacteristic
392C2#2840957711yescharacteristic
393C2#2840957911yescharacteristic
394C2#2840958111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#388×C227C2
prime radicalC22×C22⋊D227C22
hypercenter#388D227
hypocenterC227#2860
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#20300283C2semidirect
2. C2×#20300281C2semidirect
6. #28408929C2semidirect
7. #28408929C2semidirect
8. #28408927C2semidirect
9. #28408925C2semidirect
10. #28408924C2semidirect
11. #28408923C2semidirect
12. #28408921C2semidirect
13. #28408920C2semidirect
14. C42×Dic227C2semidirect
15. #388×C227C2semidirect
86. C4×Dic227D4semidirect
187. Dic227#126semidirect
197. C227#2860semidirect
212. #388D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002832
C2×#203002812
#284089322
#284089292
#284089272
#284089252
#284089242
#284089232
#284089212
#284089202
C42×Dic2272
#388×C2272
#2860227