Finite Group Databasebeta

← #3271×C227 #44397227

Subgroups — #3271×C227 #44397227

Order 29056.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#137×C227C211yesmaximal, characteristic, complemented
2C2×#135×C227C211yesmaximal, characteristic, complemented
3#550×C227C211yesmaximal, characteristic, complemented
4#548×C227C211yesmaximal, characteristic, complemented
5#496×C227C211yesmaximal, characteristic, complemented
6#469×C227C211yesmaximal, characteristic, complemented
7#465×C227C211yesmaximal, characteristic, complemented
8#464×C227C211yesmaximal, characteristic, complemented
9#460×C227C211yesmaximal, characteristic, complemented
10#452×C227C211yesmaximal, characteristic, complemented
11#449×C227C211yesmaximal, characteristic, complemented
12#449×C227C211yesmaximal, characteristic, complemented
13#448×C227C211yesmaximal, characteristic, complemented
14#441×C227C211yesmaximal, characteristic, complemented
15#435×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227C2211yescharacteristic
17C2×#41×C227C2211yescharacteristic
18C2×#41×C227C2211yescharacteristic
19C22×Q8×C227C2211yescharacteristic
20#137×C227—24no—
21#135×C227—24no—
22C2×Q16×C227C2211yescharacteristic
23C2×Q16×C227C2211yescharacteristic
24C2×SD16×C227C2211yescharacteristic
25C2×SD16×C227C2211yescharacteristic
26C2×SD16×C227C2211yescharacteristic
27C2×SD16×C227C2211yescharacteristic
28C2×SD16×C227C2211yescharacteristic
29C2×D8×C227C2211yescharacteristic
30C2×D8×C227C2211yescharacteristic
31C2×M16×C227C2211yescharacteristic
32C22×C8×C227C2211yescharacteristic
33#127×C227—21no—
34#127×C227C2211yescharacteristic
35#124×C227—21no—
36#124×C227C2211yescharacteristic
37C22⋊Q8×C227—21no—
38C22⋊Q8×C227C2211yescharacteristic
39#121×C227—21no—
40#121×C227—21no—
41#121×C227—21no—
42#121×C227C2211yescharacteristic
43#121×C227C2211yescharacteristic
44#121×C227C2211yescharacteristic
45C4×Q8×C227C2211yescharacteristic
46C4×Q8×C227C2211yescharacteristic
47C4×D4×C227—21no—
48C4×D4×C227—21no—
49C4×D4×C227C2211yescharacteristic
50#106×C227C2211yescharacteristic
51C4⋊C8×C227C2211yescharacteristic
52#103×C227C2211yescharacteristic
53#103×C227C2211yescharacteristic
54#103×C227C2211yescharacteristic
55#102×C227C2211yescharacteristic
56#102×C227C2211yescharacteristic
57#102×C227C2211yescharacteristic
58C22⋊C8×C227C2211yescharacteristic
59C22⋊C8×C227C2211yescharacteristic
60#97×C227C2211yescharacteristic
61#41×C227—22no—
62#41×C227—22no—
63#41×C227—22no—
64#41×C227—41no—
65#41×C227—41no—
66#41×C227—41no—
67C2×Q8×C227—21no—
68C2×Q8×C227—22no—
69C2×Q8×C227—41no—
70C2×Q8×C227C2311yescharacteristic
71C2×Q8×C227C2311yescharacteristic
72C2×Q8×C227C2311yescharacteristic
73C2×D4×C227—21no—
74C2×D4×C227—21no—
75C2×D4×C227—21no—
76C2×D4×C227—21no—
77C2×D4×C227—21no—
78C2×D4×C227—21no—
79C2×D4×C227—21no—
80C2×D4×C227—21no—
81C2×D4×C227D411yescharacteristic
82C2×D4×C227C2311yescharacteristic
83C2×D4×C227C2311yescharacteristic
84C2×D4×C227C2311yescharacteristic
85C22×C4×C227—21no—
86C22×C4×C227—21no—
87C22×C4×C227—21no—
88C22×C4×C227—21no—
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91Q16×C227—22no—
92Q16×C227D414yescomplemented
93SD16×C227—22no—
94SD16×C227—22no—
95SD16×C227—22no—
96SD16×C227—22no—
97SD16×C227—41no—
98D8×C227—22no—
99D8×C227—41no—
100M16×C227—22no—
101C2×C8×C227—22no—
102C2×C8×C227C2311yescharacteristic
103C2×C8×C227C2311yescharacteristic
104C2×C8×C227C2311yescharacteristic
105C2×C8×C227C2311yescharacteristic
106C4⋊C4×C227—21no—
107C4⋊C4×C227—21no—
108C4⋊C4×C227—21no—
109C4⋊C4×C227D411yescharacteristic
110C4⋊C4×C227C2311yescharacteristic
111C4⋊C4×C227C2311yescharacteristic
112C22⋊C4×C227—21no—
113C22⋊C4×C227—21no—
114C22⋊C4×C227—21no—
115C22⋊C4×C227—21no—
116C22⋊C4×C227—21no—
117C22⋊C4×C227—21no—
118C22⋊C4×C227D411yescharacteristic
119C22⋊C4×C227D411yescharacteristic
120C42×C227—21no—
121C42×C227—21no—
122C42×C227C2311yescharacteristic
123C23×C227—21no—
124C23×C227—21no—
125C23×C227—21no—
126C23×C227C2×D411yescharacteristic
127C23×C227C2×D411yescharacteristic
128Q8×C227—21no—
129Q8×C227—21no—
130Q8×C227—21no—
131Q8×C227—21no—
132Q8×C227—42no—
133Q8×C227C2×D412yes—
134Q8×C227C2×D412yes—
135D4×C227—21no—
136D4×C227—21no—
137D4×C227—21no—
138D4×C227—21no—
139D4×C227—21no—
140D4×C227—21no—
141D4×C227—41no—
142D4×C227—42no—
143D4×C227—42no—
144D4×C227—42no—
145D4×C227—42no—
146D4×C227—42no—
147D4×C227—81no—
148D4×C227—81no—
149D4×C227—81no—
150C2×C4×C227—21no—
151C2×C4×C227—21no—
152C2×C4×C227—21no—
153C2×C4×C227—21no—
154C2×C4×C227—21no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227—21no—
158C2×C4×C227—21no—
159C2×C4×C227—42no—
160C2×C4×C227—42no—
161C2×C4×C227—42no—
162C2×C4×C227—42no—
163C2×C4×C227C2×D411yescharacteristic
164C2×C4×C227C2×D411yescharacteristic
165C2×C4×C227C2×D411yescharacteristic
166C2×C4×C227C2×D411yescharacteristic
167C2×C4×C227C2411yescharacteristic
168C8×C227—21no—
169C8×C227—21no—
170C8×C227—21no—
171C8×C227C2×D412yes—
172C22×C227—21no—
173C22×C227—21no—
174C22×C227—21no—
175C22×C227—21no—
176C22×C227—21no—
177C22×C227—21no—
178C22×C227—41no—
179C22×C227—41no—
180C22×C227—41no—
181C22×C227—41no—
182C22×C227—41no—
183C22×C227—41no—
184C22×C227—41no—
185C22×C227—41no—
186C22×C227—41no—
187C22×C227C22×D411yescharacteristic
188C4×C227—21no—
189C4×C227—21no—
190C4×C227—22no—
191C4×C227—22no—
192C4×C227—21no—
193C4×C227—21no—
194C4×C227—41no—
195C4×C227—41no—
196C4×C227—41no—
197C4×C227C22×D411yescharacteristic
198C4×C227H(5,𝔽2)11yescharacteristic
199C2×C227—41no—
200C2×C227—41no—
201C2×C227—81no—
202C2×C227—81no—
203C2×C227—81no—
204C2×C227D4211yescharacteristic
205C2×C227#57511yescharacteristic
206C2×C227#57711yescharacteristic
207C227#327111yescharacteristic, direct product factor, Sylow
208#3271C22711yesmaximal, characteristic, direct product factor, Sylow
209C2×#137C2×C22711yescharacteristic, complemented
210C2×#135C2×C22711yescharacteristic, complemented
211#550C2×C22711yescharacteristic, complemented
212#548C2×C22711yescharacteristic, complemented
213#496C2×C22711yescharacteristic, complemented
214#469C2×C22711yescharacteristic, complemented
215#465C2×C22711yescharacteristic, complemented
216#464C2×C22711yescharacteristic, complemented
217#460C2×C22711yescharacteristic, complemented
218#452C2×C22711yescharacteristic, complemented
219#449C2×C22711yescharacteristic, complemented
220#449C2×C22711yescharacteristic, complemented
221#448C2×C22711yescharacteristic, complemented
222#441C2×C22711yescharacteristic, complemented
223#435C2×C22711yescharacteristic, complemented
224C2×#41C22×C22711yescharacteristic
225C2×#41C22×C22711yescharacteristic
226C2×#41C22×C22711yescharacteristic
227C22×Q8C22×C22711yescharacteristic
228#137—24no—
229#135—24no—
230C2×Q16C22×C22711yescharacteristic
231C2×Q16C22×C22711yescharacteristic
232C2×SD16C22×C22711yescharacteristic
233C2×SD16C22×C22711yescharacteristic
234C2×SD16C22×C22711yescharacteristic
235C2×SD16C22×C22711yescharacteristic
236C2×SD16C22×C22711yescharacteristic
237C2×D8C22×C22711yescharacteristic
238C2×D8C22×C22711yescharacteristic
239C2×M16C22×C22711yescharacteristic
240C22×C8C22×C22711yescharacteristic
241#127—21no—
242#127C22×C22711yescharacteristic
243#124—21no—
244#124C22×C22711yescharacteristic
245C22⋊Q8—21no—
246C22⋊Q8C22×C22711yescharacteristic
247#121—21no—
248#121—21no—
249#121—21no—
250#121C22×C22711yescharacteristic
251#121C22×C22711yescharacteristic
252#121C22×C22711yescharacteristic
253C4×Q8C22×C22711yescharacteristic
254C4×Q8C22×C22711yescharacteristic
255C4×D4—21no—
256C4×D4—21no—
257C4×D4C22×C22711yescharacteristic
258#106C22×C22711yescharacteristic
259C4⋊C8C22×C22711yescharacteristic
260#103C22×C22711yescharacteristic
261#103C22×C22711yescharacteristic
262#103C22×C22711yescharacteristic
263#102C22×C22711yescharacteristic
264#102C22×C22711yescharacteristic
265#102C22×C22711yescharacteristic
266C22⋊C8C22×C22711yescharacteristic
267C22⋊C8C22×C22711yescharacteristic
268#97C22×C22711yescharacteristic
269#41—22no—
270#41—22no—
271#41—22no—
272#41—41no—
273#41—41no—
274#41—41no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C23×C22711yescharacteristic
279C2×Q8C23×C22711yescharacteristic
280C2×Q8C23×C22711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4D4×C22711yescharacteristic
290C2×D4C23×C22711yescharacteristic
291C2×D4C23×C22711yescharacteristic
292C2×D4C23×C22711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C23×C22711yescharacteristic
298C22×C4C23×C22711yescharacteristic
299Q16—22no—
300Q16D4×C22714yescomplemented
301SD16—22no—
302SD16—22no—
303SD16—22no—
304SD16—22no—
305SD16—41no—
306D8—22no—
307D8—41no—
308M16—22no—
309C2×C8—22no—
310C2×C8C23×C22711yescharacteristic
311C2×C8C23×C22711yescharacteristic
312C2×C8C23×C22711yescharacteristic
313C2×C8C23×C22711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4D4×C22711yescharacteristic
318C4⋊C4C23×C22711yescharacteristic
319C4⋊C4C23×C22711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4D4×C22711yescharacteristic
327C22⋊C4D4×C22711yescharacteristic
328C42—21no—
329C42—21no—
330C42C23×C22711yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23C2×D4×C22711yescharacteristic
335C23C2×D4×C22711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8C2×D4×C22712yes—
342Q8C2×D4×C22712yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4C2×D4×C22711yescharacteristic
372C2×C4C2×D4×C22711yescharacteristic
373C2×C4C2×D4×C22711yescharacteristic
374C2×C4C2×D4×C22711yescharacteristic
375C2×C4C24×C22711yescharacteristic
376C8—21no—
377C8—21no—
378C8—21no—
379C8C2×D4×C22712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C22×D4×C22711yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4C22×D4×C22711yescharacteristic
406C4H(5,𝔽2)×C22711yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—81no—
412C2D42×C22711yescharacteristic
413C2#575×C22711yescharacteristic
414C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C227 ◁ C4×D4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#137×C227C2semidirect
2. C2×#135×C227C2semidirect
3. #550×C227C2semidirect
4. #548×C227C2semidirect
5. #496×C227C2semidirect
6. #469×C227C2semidirect
7. #465×C227C2semidirect
8. #464×C227C2semidirect
9. #460×C227C2semidirect
10. #452×C227C2semidirect
11. #449×C227C2semidirect
12. #449×C227C2semidirect
13. #448×C227C2semidirect
14. #441×C227C2semidirect
15. #435×C227C2semidirect
92. Q16×C227D4semidirect
207. C227#3271direct (N × H)
208. #3271C227direct (N × H)
209. C2×#137C2×C227semidirect
210. C2×#135C2×C227semidirect
211. #550C2×C227semidirect
212. #548C2×C227semidirect
213. #496C2×C227semidirect
214. #469C2×C227semidirect
215. #465C2×C227semidirect
216. #464C2×C227semidirect
217. #460C2×C227semidirect
218. #452C2×C227semidirect
219. #449C2×C227semidirect
220. #449C2×C227semidirect
221. #448C2×C227semidirect
222. #441C2×C227semidirect
223. #435C2×C227semidirect
300. Q16D4×C227semidirect

Maximal subgroups

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

subgroupindex
C2×#137×C2272
C2×#135×C2272
#550×C2272
#548×C2272
#496×C2272
#469×C2272
#465×C2272
#464×C2272
#460×C2272
#452×C2272
#449×C2272
#448×C2272
#441×C2272
#435×C2272
#3271227