Finite Group Databasebeta

← #44409187

Subgroups — #44409187

Order 29056.

496 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×#20300310C211yesmaximal, characteristic, complemented
2C2×#20300308C211yesmaximal, characteristic, complemented
3#28409578C211yesmaximal, characteristic, complemented
4#28409500C211yesmaximal, characteristic, complemented
5#28409257C211yesmaximal, characteristic, complemented
6#28409253C211yesmaximal, characteristic, complemented
7#28409250C211yesmaximal, characteristic, complemented
8#28409246C211yesmaximal, characteristic, complemented
9#28409241C211yesmaximal, characteristic, complemented
10#28409236C211yesmaximal, characteristic, complemented
11#28409231C211yesmaximal, characteristic, complemented
12#28409226C211yesmaximal, characteristic, complemented
13C2×Dic227⋊C8C211yesmaximal, characteristic, complemented
14#28409026C211yesmaximal, characteristic, complemented
15#416×C227C211yesmaximal, characteristic, complemented
16C2×#16813884C2211yescharacteristic
17C2×#16813884C2211yescharacteristic
18#20300310—24no—
19#20300308—24no—
20C2×Dic908C2211yescharacteristic
21C2×#16813874C2211yescharacteristic
22C2×#16813873C2211yescharacteristic
23C2×#16813873C2211yescharacteristic
24C2×#16813872C2211yescharacteristic
25C2×#16813871C2211yescharacteristic
26C2×D1816C2211yescharacteristic
27C2×#16813869C2211yescharacteristic
28C22×C227⋊C8C2211yescharacteristic
29#20300286C2211yescharacteristic
30#20300284C2211yescharacteristic
31#20300277C2211yescharacteristic
32#20300275C2211yescharacteristic
33#20300260—21no—
34#20300260—21no—
35#20300253C2211yescharacteristic
36#20300253C2211yescharacteristic
37#20300250C2211yescharacteristic
38#20300250C2211yescharacteristic
39C4×C22⋊D227—21no—
40C4×C22⋊D227—21no—
41(C22⋊D227)⋊C4—21no—
42(C22⋊D227)⋊C4—21no—
43#20300245—21no—
44C2×C4×Dic227C2211yescharacteristic
45C42×D227—21no—
46Dic227⋊C8C2211yescharacteristic
47Dic227⋊C8C2211yescharacteristic
48Dic227⋊C8C2211yescharacteristic
49Dic227⋊C8C2211yescharacteristic
50#20300211C2211yescharacteristic
51#20300211C2211yescharacteristic
52#20300208C2211yescharacteristic
53#20300208C2211yescharacteristic
54C2×#41×C227C2211yescharacteristic
55C22×C8×C227C2211yescharacteristic
56#117×C227C2211yescharacteristic
57#103×C227C2211yescharacteristic
58#103×C227C2211yescharacteristic
59#102×C227C2211yescharacteristic
60#102×C227C2211yescharacteristic
61#16813884—22no—
62#16813884—22no—
63#16813884—22no—
64#16813884D414yescomplemented
65C2×Dic454C2311yescharacteristic
66C2×Dic454C2311yescharacteristic
67C2×D908C2311yescharacteristic
68C2×D908C2311yescharacteristic
69C2×C22⋊D227—21no—
70C2×C22⋊D227—21no—
71C2×C22⋊D227—21no—
72C2×C22⋊D227—21no—
73C22×Dic227D411yescharacteristic
74C22×Dic227D411yescharacteristic
75C2×C4×D227—21no—
76C2×C4×D227—21no—
77C2×C4×D227—21no—
78C2×C4×D227—21no—
79Dic908—22no—
80#16813874—22no—
81#16813873—22no—
82#16813873—22no—
83#16813872—22no—
84#16813871—22no—
85D1816—22no—
86#16813869—22no—
87C2×C227⋊C8—22no—
88C2×C227⋊C8C2311yescharacteristic
89C2×C227⋊C8C2311yescharacteristic
90Dic227⋊C4—21no—
91Dic227⋊C4—21no—
92Dic227⋊C4—21no—
93Dic227⋊C4—21no—
94C22⋊Dic227—21no—
95C22⋊Dic227—21no—
96(C2×D227)⋊C4—21no—
97(C2×D227)⋊C4—21no—
98(C2×D227)⋊C4—21no—
99(C2×D227)⋊C4—21no—
100C4×Dic227—21no—
101C4×Dic227—21no—
102C4×Dic227D411yescharacteristic
103C4×Dic227D411yescharacteristic
104C4×Dic227C2311yescharacteristic
105C4×Dic227C2311yescharacteristic
106#41×C227—22no—
107#41×C227—22no—
108C2×Q8×C227C2311yescharacteristic
109C2×D4×C227—21no—
110C2×D4×C227C2311yescharacteristic
111C22×C4×C227—21no—
112C22×C4×C227C2311yescharacteristic
113C2×C8×C227—22no—
114C2×C8×C227C2311yescharacteristic
115C2×C8×C227C2311yescharacteristic
116C4⋊C4×C227C2311yescharacteristic
117C4⋊C4×C227C2311yescharacteristic
118C22⋊C4×C227—21no—
119C42×C227—21no—
120C22×D227—21no—
121C22×D227—21no—
122Dic454—21no—
123Dic454—21no—
124Dic454—21no—
125Dic454C2×D412yes—
126D908—21no—
127D908—21no—
128D908—21no—
129D908C2×D412yes—
130C22⋊D227—24no—
131C22⋊D227—42no—
132C22⋊D227—42no—
133C22⋊D227—42no—
134C2×Dic227—21no—
135C2×Dic227—22no—
136C2×Dic227—21no—
137C2×Dic227—41no—
138C2×Dic227C2×D411yescharacteristic
139C2×Dic227C2×D411yescharacteristic
140C2×Dic227C2×D411yescharacteristic
141C2×Dic227C2×D411yescharacteristic
142C4×D227—24no—
143C4×D227—42no—
144C4×D227—42no—
145C4×D227—42no—
146C227⋊C8—21no—
147C227⋊C8—21no—
148C23×C227—21no—
149C23×C227C2×D411yescharacteristic
150Q8×C227—21no—
151Q8×C227—21no—
152D4×C227—21no—
153D4×C227—21no—
154D4×C227—42no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227—21no—
158C2×C4×C227—42no—
159C2×C4×C227C2×D411yescharacteristic
160C2×C4×C227C2×D412yes—
161C2×C4×C227#4112yes—
162C2×C4×C227C2411yescharacteristic
163C8×C227—21no—
164C8×C227—21no—
165C22×C227—21no—
166C22×C227—21no—
167C22×C227—41no—
168C22×C227—41no—
169C22×C227—41no—
170C22×C227#12112yes—
171C22×C227C22×D411yescharacteristic
172C2×D227—22no—
173C2×D227—41no—
174C2×D227—41no—
175C2×D227—41no—
176C2×D227—41no—
177C2×D227—41no—
178C4×C227—41no—
179C4×C227—41no—
180C4×C227—41no—
181C4×C227#12112yes—
182C4×C227C22×D411yescharacteristic
183C4×C227C2×#4111yescharacteristic
184Dic227—22no—
185Dic227—21no—
186Dic227—21no—
187Dic227—21no—
188Dic227—41no—
189Dic227#13512yescomplemented
190C2×C227—22no—
191C2×C227—81no—
192C2×C227C2×#12111yescharacteristic
193C2×C227C2×#13511yescharacteristic
194C2×C227#57511yescharacteristic
195D227—42no—
196D227—81no—
197C227#303511yescharacteristic, complemented, Sylow
198#3035—2271nomaximal, Sylow
199C2×#135—2271no—
200C2×#135—2271no—
201#532—2271no—
202C4×#41—2271no—
203#464—2271no—
204#464—2271no—
205#463—2271no—
206#463—2271no—
207#462—2271no—
208#461—2271no—
209#460—2271no—
210#459—2271no—
211C2×C4⋊C8—2271no—
212#416—2271no—
213#416D22711yescharacteristic, complemented
214C2×#41—2271no—
215C2×#41—2271no—
216C2×#41C2×D22711yescharacteristic
217#135—4544no—
218#135—4544no—
219C2×Q16—2271no—
220C2×Q16—2271no—
221C2×SD16—2271no—
222C2×SD16—2271no—
223C2×SD16—2271no—
224C2×SD16—2271no—
225C2×D8—2271no—
226C2×D8—2271no—
227C22×C8—2271no—
228C22×C8C2×D22711yescharacteristic
229C4⋊Q8—2271no—
230#127—2271no—
231#124—2271no—
232#124—2271no—
233#121—4541no—
234#121—4541no—
235C4×Q8—2271no—
236C4×Q8—2271no—
237C4×D4—4541no—
238C4×D4—4541no—
239C4×D4—4541no—
240C4×D4—4541no—
241C4×D4—2271no—
242C4×D4—2271no—
243#117—4541no—
244#117C2×D22711yescharacteristic
245C2×C42—4541no—
246C2×C42—2271no—
247C4⋊C8—2271no—
248C4⋊C8—2271no—
249C4⋊C8—2271no—
250C4⋊C8—2271no—
251#103—2271no—
252#103—2271no—
253#103C2×D22711yescharacteristic
254#103C2×D22711yescharacteristic
255#102—2271no—
256#102—2271no—
257#102C2×D22711yescharacteristic
258#102C2×D22711yescharacteristic
259#41—22no—
260#41—22no—
261#41—4542no—
262#41—4542no—
263#41—4542no—
264#41—2274no—
265C2×Q8—2271no—
266C2×Q8—2271no—
267C2×Q8C22×D22711yescharacteristic
268C2×D4—21no—
269C2×D4—4541no—
270C2×D4—4541no—
271C2×D4—4541no—
272C2×D4—4541no—
273C2×D4—2271no—
274C2×D4—2271no—
275C2×D4C22×D22711yescharacteristic
276C22×C4—21no—
277C22×C4—4541no—
278C22×C4—4541no—
279C22×C4—4541no—
280C22×C4—4541no—
281C22×C4—2271no—
282C22×C4—2271no—
283C22×C4C22×D22711yescharacteristic
284Q16—4542no—
285Q16—4542no—
286SD16—4542no—
287SD16—4542no—
288SD16—4542no—
289SD16—4542no—
290D8—4542no—
291D8—4542no—
292C2×C8—22no—
293C2×C8—4542no—
294C2×C8—2271no—
295C2×C8—2271no—
296C2×C8C22×D22711yescharacteristic
297C2×C8C22×D22711yescharacteristic
298C4⋊C4—4541no—
299C4⋊C4—4541no—
300C4⋊C4—4541no—
301C4⋊C4—4541no—
302C4⋊C4C22×D22711yescharacteristic
303C4⋊C4C22×D22711yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—4541no—
306C22⋊C4—4541no—
307C22⋊C4—4541no—
308C22⋊C4—4541no—
309C22⋊C4—4541no—
310C22⋊C4—4541no—
311C42—21no—
312C42—4541no—
313C42—4541no—
314C42—2271no—
315C42—2271no—
316C42—2271no—
317C42—2271no—
318C23—21no—
319C23—4541no—
320C23—4541no—
321C23D4×D22711yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—4541no—
325Q8—4541no—
326Q8—4541no—
327Q8—2272no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—9082no—
332D4—9082no—
333D4—9082no—
334D4—4544no—
335D4—4541no—
336D4—4541no—
337D4—4541no—
338D4—2272no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—9082no—
344C2×C4—9082no—
345C2×C4—9082no—
346C2×C4—9081no—
347C2×C4—4541no—
348C2×C4—4541no—
349C2×C4—4542no—
350C2×C4—4544no—
351C2×C4—2271no—
352C2×C4—2271no—
353C2×C4—2271no—
354C2×C4—2271no—
355C2×C4D4×D22712yes—
356C2×C4D4×D22711yescharacteristic
357C2×C4#1681388412yes—
358C2×C4C23×D22711yescharacteristic
359C8—21no—
360C8—21no—
361C8—4541no—
362C8—4541no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—9081no—
369C22—9081no—
370C22—9081no—
371C22—9081no—
372C22—9081no—
373C22—4542no—
374C22#2030026112yes—
375C22C2×D4×D22711yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—9081no—
380C4—4542no—
381C4—4541no—
382C4—4541no—
383C4—4541no—
384C4—2272no—
385C4#2030026112yes—
386C4C2×D4×D22711yescharacteristic
387C4C2×#1681388411yescharacteristic
388C2—22no—
389C2—81no—
390C2—18161no—
391C2—9082no—
392C2C2×#2030026111yescharacteristic
393C2#135×D22711yescharacteristic
394C2#2840982911yescharacteristic
subgroupG/H
centerC2×C4D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#416×C227C2
socleC22×C227C22×D4
hypercenter#416D227
hypocenterC227#3035
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#20300310C2semidirect
2. C2×#20300308C2semidirect
3. #28409578C2semidirect
4. #28409500C2semidirect
5. #28409257C2semidirect
6. #28409253C2semidirect
7. #28409250C2semidirect
8. #28409246C2semidirect
9. #28409241C2semidirect
10. #28409236C2semidirect
11. #28409231C2semidirect
12. #28409226C2semidirect
13. C2×Dic227⋊C8C2semidirect
14. #28409026C2semidirect
15. #416×C227C2semidirect
64. #16813884D4semidirect
189. Dic227#135semidirect
197. C227#3035semidirect
213. #416D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003102
C2×#203003082
#284095782
#284095002
#284092572
#284092532
#284092502
#284092462
#284092412
#284092362
#284092312
#284092262
C2×Dic227⋊C82
#284090262
#416×C2272
#3035227