Finite Group Databasebeta

← #2716×C211 #41010336

Subgroups — #2716×C211 #41010336

Order 27008.

450 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×C211C211yesmaximal, characteristic, complemented
2C2×#126×C211C211yesmaximal, characteristic
3#400×C211C211yesmaximal, characteristic, complemented
4#398×C211C211yesmaximal, characteristic, complemented
5#397×C211C211yesmaximal, characteristic
6#396×C211C211yesmaximal, characteristic
7#393×C211C211yesmaximal, characteristic
8#393×C211C211yesmaximal, characteristic
9#391×C211C211yesmaximal, characteristic, complemented
10#391×C211C211yesmaximal, characteristic, complemented
11#388×C211C211yesmaximal, characteristic
12#388×C211C211yesmaximal, characteristic
13#387×C211C211yesmaximal, characteristic, complemented
14#387×C211C211yesmaximal, characteristic, complemented
15#387×C211C211yesmaximal, characteristic, complemented
16C22×Q8×C211C2211yescharacteristic
17C22×Q8×C211C2211yescharacteristic
18C4⋊Q8×C211—24no—
19#126×C211—24no—
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C2×C4⋊C4×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C4⋊C4×C211C2211yescharacteristic
29C2×C22⋊C4×C211C2211yescharacteristic
30C2×C22⋊C4×C211C2211yescharacteristic
31C2×C22⋊C4×C211C2211yescharacteristic
32C2×C22⋊C4×C211C2211yescharacteristic
33C2×C22⋊C4×C211C2211yescharacteristic
34C2×C22⋊C4×C211C2211yescharacteristic
35C2×C22⋊C4×C211C2211yescharacteristic
36C2×C42×C211C2211yescharacteristic
37C2×C42×C211C2211yescharacteristic
38C2×C42×C211C2211yescharacteristic
39#95×C211C2211yescharacteristic
40#95×C211C2211yescharacteristic
41#95×C211C2211yescharacteristic
42#95×C211C2211yescharacteristic
43#95×C211C2211yescharacteristic
44#95×C211C2211yescharacteristic
45#95×C211C2211yescharacteristic
46#95×C211C2211yescharacteristic
47#95×C211C2211yescharacteristic
48#95×C211C2211yescharacteristic
49#95×C211C2211yescharacteristic
50#95×C211C2211yescharacteristic
51#95×C211C2211yescharacteristic
52#95×C211C2211yescharacteristic
53C24×C211C2311yescharacteristic
54C2×Q8×C211—22no—
55C2×Q8×C211—22no—
56C2×Q8×C211—22no—
57C2×Q8×C211—22no—
58C2×Q8×C211—41no—
59C2×Q8×C211—41no—
60C22×C4×C211C2311yescharacteristic
61C22×C4×C211C2311yescharacteristic
62C22×C4×C211C2311yescharacteristic
63C22×C4×C211C2311yescharacteristic
64C22×C4×C211C2311yescharacteristic
65C22×C4×C211C2311yescharacteristic
66C22×C4×C211C2311yescharacteristic
67C22×C4×C211C2311yescharacteristic
68C22×C4×C211C2311yescharacteristic
69C22×C4×C211C2311yescharacteristic
70C22×C4×C211C2311yescharacteristic
71C22×C4×C211C2311yescharacteristic
72C22×C4×C211C2311yescharacteristic
73C22×C4×C211C2311yescharacteristic
74C4⋊C4×C211—22no—
75C4⋊C4×C211—22no—
76C4⋊C4×C211—22no—
77C4⋊C4×C211—22no—
78C4⋊C4×C211—22no—
79C4⋊C4×C211—22no—
80C4⋊C4×C211—22no—
81C4⋊C4×C211—41no—
82C4⋊C4×C211D414yes—
83C22⋊C4×C211—22no—
84C22⋊C4×C211—22no—
85C22⋊C4×C211—22no—
86C22⋊C4×C211—22no—
87C22⋊C4×C211—22no—
88C22⋊C4×C211—41no—
89C22⋊C4×C211—41no—
90C42×C211—22no—
91C42×C211—22no—
92C42×C211—41no—
93C23×C211—21no—
94C23×C211—21no—
95C23×C211—21no—
96C23×C211—21no—
97C23×C211—21no—
98C23×C211—21no—
99C23×C211—21no—
100C23×C211C2411yescharacteristic
101Q8×C211—44no—
102Q8×C211—44no—
103C2×C4×C211—21no—
104C2×C4×C211—21no—
105C2×C4×C211—21no—
106C2×C4×C211—21no—
107C2×C4×C211—21no—
108C2×C4×C211—21no—
109C2×C4×C211—21no—
110C2×C4×C211—21no—
111C2×C4×C211—21no—
112C2×C4×C211—21no—
113C2×C4×C211—21no—
114C2×C4×C211—21no—
115C2×C4×C211—21no—
116C2×C4×C211—21no—
117C2×C4×C211—21no—
118C2×C4×C211—21no—
119C2×C4×C211—21no—
120C2×C4×C211—21no—
121C2×C4×C211—21no—
122C2×C4×C211—21no—
123C2×C4×C211—21no—
124C2×C4×C211—21no—
125C2×C4×C211—21no—
126C2×C4×C211—21no—
127C2×C4×C211—21no—
128C2×C4×C211—21no—
129C2×C4×C211—21no—
130C2×C4×C211—21no—
131C2×C4×C211—21no—
132C2×C4×C211—21no—
133C2×C4×C211—21no—
134C2×C4×C211—21no—
135C2×C4×C211—21no—
136C2×C4×C211—21no—
137C2×C4×C211—21no—
138C2×C4×C211—21no—
139C2×C4×C211—21no—
140C2×C4×C211C2×D412yes—
141C2×C4×C211C2×D412yes—
142C2×C4×C211C2×D412yes—
143C2×C4×C211#4112yes—
144C2×C4×C211#4112yes—
145C22×C211—41no—
146C22×C211—41no—
147C22×C211—41no—
148C22×C211—41no—
149C22×C211—41no—
150C22×C211—41no—
151C22×C211—41no—
152C22×C211C22×D411yescharacteristic
153C22×C211C2×#4111yescharacteristic
154C22×C211C2×#4111yescharacteristic
155C22×C211H(5,𝔽2)11yescharacteristic
156C22×C2112−1+411yescharacteristic
157C22×C2112−1+411yescharacteristic
158C22×C2112−1+411yescharacteristic
159C4×C211—22no—
160C4×C211—22no—
161C4×C211—22no—
162C4×C211—22no—
163C4×C211—22no—
164C4×C211—41no—
165C4×C211—41no—
166C4×C211—41no—
167C4×C211—41no—
168C4×C211—41no—
169C4×C211—41no—
170C4×C211—41no—
171C4×C211—41no—
172C4×C211—41no—
173C2×C211—81no—
174C2×C211#53611yescharacteristic
175C2×C211#54111yescharacteristic
176C2×C211#54211yescharacteristic
177C2×C211#54611yescharacteristic
178C2×C211#54811yescharacteristic
179C2×C211#55611yescharacteristic
180C2×C211#56311yescharacteristic
181C211#271611yescharacteristic, direct product factor, Sylow
182#2716C21111yesmaximal, characteristic, direct product factor, Sylow
183C2×C4⋊Q8C2×C21111yescharacteristic, complemented
184C2×#126C2×C21111yescharacteristic
185#400C2×C21111yescharacteristic, complemented
186#398C2×C21111yescharacteristic, complemented
187#397C2×C21111yescharacteristic
188#396C2×C21111yescharacteristic
189#393C2×C21111yescharacteristic
190#393C2×C21111yescharacteristic
191#391C2×C21111yescharacteristic, complemented
192#391C2×C21111yescharacteristic, complemented
193#388C2×C21111yescharacteristic
194#388C2×C21111yescharacteristic
195#387C2×C21111yescharacteristic, complemented
196#387C2×C21111yescharacteristic, complemented
197#387C2×C21111yescharacteristic, complemented
198C22×Q8C22×C21111yescharacteristic
199C22×Q8C22×C21111yescharacteristic
200C4⋊Q8—24no—
201#126—24no—
202C2×C4⋊C4C22×C21111yescharacteristic
203C2×C4⋊C4C22×C21111yescharacteristic
204C2×C4⋊C4C22×C21111yescharacteristic
205C2×C4⋊C4C22×C21111yescharacteristic
206C2×C4⋊C4C22×C21111yescharacteristic
207C2×C4⋊C4C22×C21111yescharacteristic
208C2×C4⋊C4C22×C21111yescharacteristic
209C2×C4⋊C4C22×C21111yescharacteristic
210C2×C4⋊C4C22×C21111yescharacteristic
211C2×C22⋊C4C22×C21111yescharacteristic
212C2×C22⋊C4C22×C21111yescharacteristic
213C2×C22⋊C4C22×C21111yescharacteristic
214C2×C22⋊C4C22×C21111yescharacteristic
215C2×C22⋊C4C22×C21111yescharacteristic
216C2×C22⋊C4C22×C21111yescharacteristic
217C2×C22⋊C4C22×C21111yescharacteristic
218C2×C42C22×C21111yescharacteristic
219C2×C42C22×C21111yescharacteristic
220C2×C42C22×C21111yescharacteristic
221#95C22×C21111yescharacteristic
222#95C22×C21111yescharacteristic
223#95C22×C21111yescharacteristic
224#95C22×C21111yescharacteristic
225#95C22×C21111yescharacteristic
226#95C22×C21111yescharacteristic
227#95C22×C21111yescharacteristic
228#95C22×C21111yescharacteristic
229#95C22×C21111yescharacteristic
230#95C22×C21111yescharacteristic
231#95C22×C21111yescharacteristic
232#95C22×C21111yescharacteristic
233#95C22×C21111yescharacteristic
234#95C22×C21111yescharacteristic
235C24C23×C21111yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C22×C4C23×C21111yescharacteristic
243C22×C4C23×C21111yescharacteristic
244C22×C4C23×C21111yescharacteristic
245C22×C4C23×C21111yescharacteristic
246C22×C4C23×C21111yescharacteristic
247C22×C4C23×C21111yescharacteristic
248C22×C4C23×C21111yescharacteristic
249C22×C4C23×C21111yescharacteristic
250C22×C4C23×C21111yescharacteristic
251C22×C4C23×C21111yescharacteristic
252C22×C4C23×C21111yescharacteristic
253C22×C4C23×C21111yescharacteristic
254C22×C4C23×C21111yescharacteristic
255C22×C4C23×C21111yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4D4×C21114yes—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—22no—
273C42—22no—
274C42—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23C24×C21111yescharacteristic
283Q8—44no—
284Q8—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
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×C4C2×D4×C21112yes—
323C2×C4C2×D4×C21112yes—
324C2×C4C2×D4×C21112yes—
325C2×C4#41×C21112yes—
326C2×C4#41×C21112yes—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C22×D4×C21111yescharacteristic
335C22C2×#41×C21111yescharacteristic
336C22C2×#41×C21111yescharacteristic
337C22H(5,𝔽2)×C21111yescharacteristic
338C222−1+4×C21111yescharacteristic
339C222−1+4×C21111yescharacteristic
340C222−1+4×C21111yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C2—81no—
356C2#536×C21111yescharacteristic
357C2#541×C21111yescharacteristic
358C2#542×C21111yescharacteristic
359C2#546×C21111yescharacteristic
360C2#548×C21111yescharacteristic
361C2#556×C21111yescharacteristic
362C2#563×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C24
commutator, Frattini, cosocleC23C24×C211
prime radicalC24×C211C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×C211C2semidirect
3. #400×C211C2semidirect
4. #398×C211C2semidirect
9. #391×C211C2semidirect
10. #391×C211C2semidirect
13. #387×C211C2semidirect
14. #387×C211C2semidirect
15. #387×C211C2semidirect
181. C211#2716direct (N × H)
182. #2716C211direct (N × H)
183. C2×C4⋊Q8C2×C211semidirect
185. #400C2×C211semidirect
186. #398C2×C211semidirect
191. #391C2×C211semidirect
192. #391C2×C211semidirect
195. #387C2×C211semidirect
196. #387C2×C211semidirect
197. #387C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×C2112
C2×#126×C2112
#400×C2112
#398×C2112
#397×C2112
#396×C2112
#393×C2112
#391×C2112
#388×C2112
#387×C2112
#2716211