Finite Group Databasebeta

← #3121×C229 #44420182

Subgroups — #3121×C229 #44420182

Order 29312.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#542×C229C211yesmaximal, characteristic
2#519×C229C211yesmaximal, characteristic, complemented
3#485×C229C211yesmaximal, characteristic, complemented
4#482×C229C211yesmaximal, characteristic
5#477×C229C211yesmaximal, characteristic, complemented
6#475×C229C211yesmaximal, characteristic, complemented
7#464×C229C211yesmaximal, characteristic, complemented
8#464×C229C211yesmaximal, characteristic, complemented
9#451×C229C211yesmaximal, characteristic, complemented
10#448×C229C211yesmaximal, characteristic
11#441×C229C211yesmaximal, characteristic, complemented
12#440×C229C211yesmaximal, characteristic, complemented
13C4×Q16×C229C211yesmaximal, characteristic, complemented
14C4×SD16×C229C211yesmaximal, characteristic, complemented
15#433×C229C211yesmaximal, characteristic, complemented
16C22×Q8×C229C2211yescharacteristic
17C2×Q16×C229C2211yescharacteristic
18C2×Q16×C229C2211yescharacteristic
19C2×SD16×C229C2211yescharacteristic
20C2×SD16×C229C2211yescharacteristic
21C4⋊Q8×C229C2211yescharacteristic
22#126×C229—21no—
23#124×C229—21no—
24#124×C229C2211yescharacteristic
25#123×C229—21no—
26C22⋊Q8×C229—21no—
27C22⋊Q8×C229C2211yescharacteristic
28#121×C229C2211yescharacteristic
29C4×Q8×C229—21no—
30C4×Q8×C229—21no—
31C4×Q8×C229C2211yescharacteristic
32C4×Q8×C229C2211yescharacteristic
33C4×Q8×C229C2211yescharacteristic
34C4×Q8×C229C2211yescharacteristic
35C4×Q8×C229C2211yescharacteristic
36C4×D4×C229C2211yescharacteristic
37#117×C229—21no—
38#117×C229—21no—
39#117×C229C2211yescharacteristic
40#117×C229C2211yescharacteristic
41#107×C229C2211yescharacteristic
42#107×C229C2211yescharacteristic
43#106×C229C2211yescharacteristic
44#106×C229C2211yescharacteristic
45C4⋊C8×C229C2211yescharacteristic
46C4⋊C8×C229C2211yescharacteristic
47#103×C229C2211yescharacteristic
48#103×C229C2211yescharacteristic
49#103×C229C2211yescharacteristic
50#103×C229C2211yescharacteristic
51#103×C229C2211yescharacteristic
52#103×C229C2211yescharacteristic
53#102×C229C2211yescharacteristic
54#102×C229C2211yescharacteristic
55C22⋊C8×C229C2211yescharacteristic
56C22⋊C8×C229C2211yescharacteristic
57#97×C229C2211yescharacteristic
58C4×C8×C229C2211yescharacteristic
59C2×Q8×C229—22no—
60C2×Q8×C229—22no—
61C2×Q8×C229—21no—
62C2×Q8×C229—21no—
63C2×Q8×C229C2311yescharacteristic
64C2×Q8×C229C2311yescharacteristic
65C2×Q8×C229C2311yescharacteristic
66C2×D4×C229—21no—
67C2×D4×C229C2311yescharacteristic
68C22×C4×C229—21no—
69C22×C4×C229—21no—
70C22×C4×C229C2311yescharacteristic
71Q16×C229—22no—
72Q16×C229—22no—
73SD16×C229—22no—
74SD16×C229—22no—
75C2×C8×C229C2311yescharacteristic
76C2×C8×C229C2311yescharacteristic
77C2×C8×C229C2311yescharacteristic
78C2×C8×C229C2311yescharacteristic
79C4⋊C4×C229—21no—
80C4⋊C4×C229—21no—
81C4⋊C4×C229—21no—
82C4⋊C4×C229—21no—
83C4⋊C4×C229—21no—
84C4⋊C4×C229—21no—
85C4⋊C4×C229—21no—
86C4⋊C4×C229D411yescharacteristic
87C4⋊C4×C229D411yescharacteristic
88C4⋊C4×C229C2311yescharacteristic
89C4⋊C4×C229C2311yescharacteristic
90C4⋊C4×C229C2311yescharacteristic
91C4⋊C4×C229C2311yescharacteristic
92C22⋊C4×C229—21no—
93C22⋊C4×C229—21no—
94C22⋊C4×C229—21no—
95C22⋊C4×C229—21no—
96C22⋊C4×C229—21no—
97C22⋊C4×C229—21no—
98C22⋊C4×C229D411yescharacteristic
99C22⋊C4×C229D411yescharacteristic
100C42×C229—21no—
101C42×C229—21no—
102C42×C229—21no—
103C42×C229—21no—
104C42×C229—21no—
105C42×C229—21no—
106C42×C229C2311yescharacteristic
107C42×C229C2311yescharacteristic
108C23×C229—21no—
109C23×C229C2×D411yescharacteristic
110Q8×C229—24no—
111Q8×C229—21no—
112Q8×C229—21no—
113Q8×C229—21no—
114Q8×C229—21no—
115Q8×C229—81no—
116Q8×C229#4112yes—
117Q8×C229#4112yes—
118D4×C229—21no—
119D4×C229—21no—
120D4×C229—81no—
121C2×C4×C229—24no—
122C2×C4×C229—21no—
123C2×C4×C229—21no—
124C2×C4×C229—21no—
125C2×C4×C229—21no—
126C2×C4×C229—21no—
127C2×C4×C229—21no—
128C2×C4×C229—21no—
129C2×C4×C229—21no—
130C2×C4×C229—21no—
131C2×C4×C229—81no—
132C2×C4×C229C2×D411yescharacteristic
133C2×C4×C229C2×D411yescharacteristic
134C2×C4×C229C2×D411yescharacteristic
135C2×C4×C229C2×D411yescharacteristic
136C2×C4×C229C2×D411yescharacteristic
137C2×C4×C229C2411yescharacteristic
138C8×C229—21no—
139C8×C229—21no—
140C8×C229—21no—
141C8×C229—21no—
142C22×C229—21no—
143C22×C229—21no—
144C22×C229—21no—
145C22×C229—41no—
146C22×C229—41no—
147C22×C229—41no—
148C22×C229C22×D411yescharacteristic
149C4×C229—22no—
150C4×C229—22no—
151C4×C229—21no—
152C4×C229—21no—
153C4×C229—21no—
154C4×C229—21no—
155C4×C229—21no—
156C4×C229—41no—
157C4×C229—41no—
158C4×C229—41no—
159C4×C229—41no—
160C4×C229—41no—
161C4×C229C2×#4111yescharacteristic
162C4×C229C2×#4111yescharacteristic
163C2×C229—41no—
164C2×C229—81no—
165C2×C229#52511yescharacteristic
166C2×C229#57711yescharacteristic
167C2×C229#57811yescharacteristic
168C229#312111yescharacteristic, direct product factor, Sylow
169#3121C22911yesmaximal, characteristic, direct product factor, Sylow
170#542C2×C22911yescharacteristic
171#519C2×C22911yescharacteristic, complemented
172#485C2×C22911yescharacteristic, complemented
173#482C2×C22911yescharacteristic
174#477C2×C22911yescharacteristic, complemented
175#475C2×C22911yescharacteristic, complemented
176#464C2×C22911yescharacteristic, complemented
177#464C2×C22911yescharacteristic, complemented
178#451C2×C22911yescharacteristic, complemented
179#448C2×C22911yescharacteristic
180#441C2×C22911yescharacteristic, complemented
181#440C2×C22911yescharacteristic, complemented
182C4×Q16C2×C22911yescharacteristic, complemented
183C4×SD16C2×C22911yescharacteristic, complemented
184#433C2×C22911yescharacteristic, complemented
185C22×Q8C22×C22911yescharacteristic
186C2×Q16C22×C22911yescharacteristic
187C2×Q16C22×C22911yescharacteristic
188C2×SD16C22×C22911yescharacteristic
189C2×SD16C22×C22911yescharacteristic
190C4⋊Q8C22×C22911yescharacteristic
191#126—21no—
192#124—21no—
193#124C22×C22911yescharacteristic
194#123—21no—
195C22⋊Q8—21no—
196C22⋊Q8C22×C22911yescharacteristic
197#121C22×C22911yescharacteristic
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8C22×C22911yescharacteristic
201C4×Q8C22×C22911yescharacteristic
202C4×Q8C22×C22911yescharacteristic
203C4×Q8C22×C22911yescharacteristic
204C4×Q8C22×C22911yescharacteristic
205C4×D4C22×C22911yescharacteristic
206#117—21no—
207#117—21no—
208#117C22×C22911yescharacteristic
209#117C22×C22911yescharacteristic
210#107C22×C22911yescharacteristic
211#107C22×C22911yescharacteristic
212#106C22×C22911yescharacteristic
213#106C22×C22911yescharacteristic
214C4⋊C8C22×C22911yescharacteristic
215C4⋊C8C22×C22911yescharacteristic
216#103C22×C22911yescharacteristic
217#103C22×C22911yescharacteristic
218#103C22×C22911yescharacteristic
219#103C22×C22911yescharacteristic
220#103C22×C22911yescharacteristic
221#103C22×C22911yescharacteristic
222#102C22×C22911yescharacteristic
223#102C22×C22911yescharacteristic
224C22⋊C8C22×C22911yescharacteristic
225C22⋊C8C22×C22911yescharacteristic
226#97C22×C22911yescharacteristic
227C4×C8C22×C22911yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8C23×C22911yescharacteristic
233C2×Q8C23×C22911yescharacteristic
234C2×Q8C23×C22911yescharacteristic
235C2×D4—21no—
236C2×D4C23×C22911yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C22911yescharacteristic
240Q16—22no—
241Q16—22no—
242SD16—22no—
243SD16—22no—
244C2×C8C23×C22911yescharacteristic
245C2×C8C23×C22911yescharacteristic
246C2×C8C23×C22911yescharacteristic
247C2×C8C23×C22911yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4D4×C22911yescharacteristic
256C4⋊C4D4×C22911yescharacteristic
257C4⋊C4C23×C22911yescharacteristic
258C4⋊C4C23×C22911yescharacteristic
259C4⋊C4C23×C22911yescharacteristic
260C4⋊C4C23×C22911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4D4×C22911yescharacteristic
268C22⋊C4D4×C22911yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42C23×C22911yescharacteristic
276C42C23×C22911yescharacteristic
277C23—21no—
278C23C2×D4×C22911yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—81no—
285Q8#41×C22912yes—
286Q8#41×C22912yes—
287D4—21no—
288D4—21no—
289D4—81no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—81no—
301C2×C4C2×D4×C22911yescharacteristic
302C2×C4C2×D4×C22911yescharacteristic
303C2×C4C2×D4×C22911yescharacteristic
304C2×C4C2×D4×C22911yescharacteristic
305C2×C4C2×D4×C22911yescharacteristic
306C2×C4C24×C22911yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C22×D4×C22911yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C2×#41×C22911yescharacteristic
331C4C2×#41×C22911yescharacteristic
332C2—41no—
333C2—81no—
334C2#525×C22911yescharacteristic
335C2#577×C22911yescharacteristic
336C2#578×C22911yescharacteristic
subgroupG/H
center, socleC22×C229C22×D4
commutator, Frattini, cosocleC2×C4C24×C229
prime radical#121×C229C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C229 ◁ #117×C229 ◁ G

Complements

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

NH ≅ G/Ntype
2. #519×C229C2semidirect
3. #485×C229C2semidirect
5. #477×C229C2semidirect
6. #475×C229C2semidirect
7. #464×C229C2semidirect
8. #464×C229C2semidirect
9. #451×C229C2semidirect
11. #441×C229C2semidirect
12. #440×C229C2semidirect
13. C4×Q16×C229C2semidirect
14. C4×SD16×C229C2semidirect
15. #433×C229C2semidirect
168. C229#3121direct (N × H)
169. #3121C229direct (N × H)
171. #519C2×C229semidirect
172. #485C2×C229semidirect
174. #477C2×C229semidirect
175. #475C2×C229semidirect
176. #464C2×C229semidirect
177. #464C2×C229semidirect
178. #451C2×C229semidirect
180. #441C2×C229semidirect
181. #440C2×C229semidirect
182. C4×Q16C2×C229semidirect
183. C4×SD16C2×C229semidirect
184. #433C2×C229semidirect

Maximal subgroups

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

subgroupindex
#542×C2292
#519×C2292
#485×C2292
#482×C2292
#477×C2292
#475×C2292
#464×C2292
#451×C2292
#448×C2292
#441×C2292
#440×C2292
C4×Q16×C2292
C4×SD16×C2292
#433×C2292
#3121229