Finite Group Databasebeta

← #41013340

Subgroups — #41013340

Order 27008.

618 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D211C211yesmaximal, characteristic
2C2×#20269905C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213109C211yesmaximal, characteristic, complemented
6C4×(C2×D211)⋊C4C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C2×#16809963C411yescharacteristic, complemented
9C2×#16809962—21no—
10C2×#16809962C2211yescharacteristic
11C2×#16809961—21no—
12C2×#16809961C411yescharacteristic, complemented
13#41×D211—22no—
14#41×D211—22no—
15#41×D211—22no—
16#41×D211—22no—
17C2×Q8×D211C411yescharacteristic, complemented
18C2×D4×D211—21no—
19C2×D4×D211C411yescharacteristic, complemented
20C22×C4×D211—21no—
21C22×C4×D211C2211yescharacteristic
22C2×#16809946C2211yescharacteristic, complemented
23C2×(C2×D211)⋊C4—21no—
24C2×C4×Dic211C2211yescharacteristic, complemented
25#20269905—22no—
26#20269904—22no—
27#20269903—22no—
28(C2×D211)⋊C8—21no—
29(C2×D211)⋊C8—21no—
30C211⋊#95—21no—
31C2×#41×C211C2211yescharacteristic
32C2×M16×C211C2211yescharacteristic, complemented
33C2×C42×C211C2211yescharacteristic, complemented
34#104×C211—22no—
35C23×D211—21no—
36C23×D211D411yescharacteristic
37#16809963—22no—
38#16809963—22no—
39#16809962—24no—
40#16809962—24no—
41#16809962—21no—
42#16809962—21no—
43#16809962D412yes—
44#16809962D412yes—
45#16809961—24no—
46#16809961—24no—
47#16809961—22no—
48#16809961—22no—
49C2×Dic422—21no—
50C2×Dic422C2×C411yescharacteristic
51Q8×D211—22no—
52Q8×D211—22no—
53C2×D844—21no—
54C2×D844C2×C411yescharacteristic
55C2×C22⋊D211—21no—
56C2×C22⋊D211—21no—
57C2×C22⋊D211—21no—
58D4×D211—24no—
59D4×D211—24no—
60D4×D211—22no—
61D4×D211—22no—
62C22×Dic211—21no—
63C22×Dic211—21no—
64C22×Dic211D411yescharacteristic
65C2×C4×D211—24no—
66C2×C4×D211—24no—
67C2×C4×D211—21no—
68C2×C4×D211—21no—
69C2×C4×D211—21no—
70C2×C4×D211—21no—
71C2×C4×D211—21no—
72C2×C4×D211—21no—
73C2×C4×D211—21no—
74C2×C4×D211C2×C411yescharacteristic
75C2×C4×D211C2×C411yescharacteristic
76C2×C4×D211D411yescharacteristic
77C2×C4×D211D411yescharacteristic
78#16809946—21no—
79#16809946—21no—
80C2×C211⋊C8—21no—
81(C2×D211)⋊C4—24no—
82C4×Dic211—21no—
83C4×Dic211—21no—
84#41×C211—21no—
85#41×C211—21no—
86#41×C211D412yes—
87#41×C211D412yes—
88C2×Q8×C211C2×C411yescharacteristic
89C2×D4×C211—21no—
90C2×D4×C211C2×C411yescharacteristic
91C22×C4×C211—21no—
92C22×C4×C211—21no—
93C22×C4×C211C2311yescharacteristic
94M16×C211—21no—
95M16×C211—21no—
96C2×C8×C211—21no—
97C42×C211—21no—
98C42×C211—21no—
99C22×D211—24no—
100C22×D211—21no—
101C22×D211—21no—
102C22×D211—21no—
103C22×D211—22no—
104C22×D211—41no—
105C22×D211—42no—
106C22×D211—42no—
107C22×D211C22⋊C411yescharacteristic
108C22×D211C22⋊C411yescharacteristic
109Dic422—24no—
110Dic422—21no—
111Dic422C22⋊C412yescomplemented
112D844—24no—
113D844—21no—
114D844C22⋊C412yescomplemented
115C22⋊D211—24no—
116C22⋊D211—24no—
117C22⋊D211—22no—
118C22⋊D211—22no—
119C2×Dic211—24no—
120C2×Dic211—21no—
121C2×Dic211—21no—
122C2×Dic211—21no—
123C2×Dic211—22no—
124C2×Dic211—41no—
125C2×Dic211—41no—
126C2×Dic211C22⋊C411yescharacteristic
127C2×Dic211C22⋊C411yescharacteristic
128C4×D211—24no—
129C4×D211—24no—
130C4×D211—21no—
131C4×D211—21no—
132C4×D211—21no—
133C4×D211—21no—
134C4×D211—21no—
135C4×D211—21no—
136C4×D211—22no—
137C4×D211—22no—
138C4×D211—21no—
139C4×D211—21no—
140C211⋊C8—41no—
141C23×C211—21no—
142C23×C211C2×D411yescharacteristic
143Q8×C211—21no—
144Q8×C211C22⋊C412yescomplemented
145D4×C211—22no—
146D4×C211—22no—
147D4×C211—21no—
148D4×C211C22⋊C412yescomplemented
149C2×C4×C211—21no—
150C2×C4×C211—22no—
151C2×C4×C211—22no—
152C2×C4×C211—22no—
153C2×C4×C211—41no—
154C2×C4×C211—41no—
155C2×C4×C211C22×C411yescharacteristic
156C2×C4×C211C2×D411yescharacteristic
157C2×C4×C211C2×D411yescharacteristic
158C2×C4×C211C2×D411yescharacteristic
159C2×C4×C211C2×D411yescharacteristic
160C2×C4×C211C2×D411yescharacteristic
161C8×C211—41no—
162C22×C211—21no—
163C22×C211—21no—
164C22×C211—22no—
165C22×C211—41no—
166C22×C211—41no—
167C22×C211C2×C22⋊C411yescharacteristic
168C22×C211#12011yescharacteristic
169C22×C211#12011yescharacteristic
170C2×D211—21no—
171C2×D211—21no—
172C2×D211—21no—
173C2×D211—22no—
174C2×D211—21no—
175C2×D211—21no—
176C2×D211—42no—
177C2×D211—42no—
178C2×D211—48no—
179C2×D211—41no—
180C2×D211—41no—
181C2×D211#10412yescomplemented
182C4×C211—22no—
183C4×C211—42no—
184C4×C211C2×C22⋊C411yescharacteristic
185C4×C211C2×C22⋊C411yescharacteristic
186C4×C211#12011yescharacteristic
187C4×C211#12011yescharacteristic
188Dic211—21no—
189Dic211—22no—
190Dic211—21no—
191Dic211—42no—
192C2×C211—21no—
193C2×C211—21no—
194C2×C211—42no—
195C2×C211#37911yescharacteristic
196C2×C211C2×#10411yescharacteristic
197C2×C211#42111yescharacteristic
198D211—22no—
199D211—41no—
200D211—42no—
201C211#178911yescharacteristic, complemented, Sylow
202#1789—2111nomaximal, Sylow
203C22×#41—2111no—
204C2×#104—2111no—
205C2×#104—2111no—
206C2×#104—2111no—
207C2×#104D21111yescharacteristic, complemented
208#407—2111no—
209C4×C22⋊C4—2111no—
210C2×#41—4221no—
211C2×#41—4222no—
212C2×#41—4222no—
213C2×#41—4222no—
214C2×#41—4222no—
215C2×#41—4221no—
216C2×#41—2111no—
217C2×#41—2111no—
218C2×#41—2111no—
219C2×#41C2×D21111yescharacteristic
220C22×Q8—2111no—
221C22×D4—4221no—
222C22×D4—2111no—
223C23×C4—4221no—
224C23×C4—2111no—
225C2×M16—2111no—
226C2×M16C2×D21111yescharacteristic, complemented
227C2×C22⋊C4—4221no—
228C2×C42—2111no—
229C2×C42C2×D21111yescharacteristic, complemented
230#104—22no—
231#104—4222no—
232#104—4222no—
233#104—4222no—
234C22⋊C8—4221no—
235C22⋊C8—4221no—
236#95—4221no—
237C24—4221no—
238C24—2111no—
239#41—21no—
240#41—21no—
241#41—4224no—
242#41—4224no—
243#41—4224no—
244#41—4224no—
245#41—4222no—
246#41—4222no—
247#41—4222no—
248#41—4222no—
249#41—4221no—
250#41—4221no—
251#41—2112no—
252#41—2112no—
253#41C22⋊D21112yes—
254#41D84412yes—
255C2×Q8—4221no—
256C2×Q8—4222no—
257C2×Q8—4222no—
258C2×Q8—2111no—
259C2×Q8C4×D21111yescharacteristic
260C2×D4—21no—
261C2×D4—4221no—
262C2×D4—4224no—
263C2×D4—4224no—
264C2×D4—4222no—
265C2×D4—4221no—
266C2×D4—4221no—
267C2×D4—4222no—
268C2×D4—4221no—
269C2×D4—2111no—
270C2×D4C4×D21111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—4224no—
274C22×C4—4224no—
275C22×C4—4221no—
276C22×C4—4221no—
277C22×C4—4221no—
278C22×C4—4221no—
279C22×C4—4221no—
280C22×C4—4221no—
281C22×C4—4221no—
282C22×C4—4221no—
283C22×C4—4221no—
284C22×C4—2111no—
285C22×C4—2111no—
286C22×C4—2111no—
287C22×C4—2111no—
288C22×C4—2111no—
289C22×C4C22×D21111yescharacteristic
290M16—21no—
291M16—21no—
292M16—4221no—
293M16—4221no—
294C2×C8—21no—
295C2×C8—4221no—
296C22⋊C4—4224no—
297C42—21no—
298C42—21no—
299C42—4221no—
300C42—4221no—
301C23—21no—
302C23—8441no—
303C23—8442no—
304C23—8442no—
305C23—4224no—
306C23—4222no—
307C23—4221no—
308C23—4221no—
309C23—4221no—
310C23—2111no—
311C23—2111no—
312C23D4×D21111yescharacteristic
313Q8—21no—
314Q8—4224no—
315Q8—4221no—
316Q8—2112no—
317Q8(C2×D211)⋊C412yescomplemented
318D4—22no—
319D4—22no—
320D4—21no—
321D4—4224no—
322D4—4224no—
323D4—4224no—
324D4—4222no—
325D4—4222no—
326D4—4221no—
327D4—2112no—
328D4(C2×D211)⋊C412yescomplemented
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—8441no—
336C2×C4—8441no—
337C2×C4—4224no—
338C2×C4—4224no—
339C2×C4—4222no—
340C2×C4—4224no—
341C2×C4—4221no—
342C2×C4—4221no—
343C2×C4—4221no—
344C2×C4—4221no—
345C2×C4—4221no—
346C2×C4—4221no—
347C2×C4—4221no—
348C2×C4—4221no—
349C2×C4—4222no—
350C2×C4—4221no—
351C2×C4—4221no—
352C2×C4—4222no—
353C2×C4—4221no—
354C2×C4—2111no—
355C2×C4—2111no—
356C2×C4C2×C4×D21111yescharacteristic
357C2×C4D4×D21111yescharacteristic
358C2×C4D4×D21111yescharacteristic
359C2×C4D4×D21111yescharacteristic
360C2×C4C2×C22⋊D21111yescharacteristic
361C2×C4C2×D84411yescharacteristic
362C8—41no—
363C8—8441no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—8442no—
370C22—8442no—
371C22—8448no—
372C22—8441no—
373C22—8441no—
374C22—4221no—
375C22—4221no—
376C22—4221no—
377C22—4222no—
378C22—4221no—
379C22—4221no—
380C22—2112no—
381C22C22⋊C4×D21111yescharacteristic
382C22#2026994511yescharacteristic
383C22#2026994711yescharacteristic
384C4—22no—
385C4—42no—
386C4—8442no—
387C4—4221no—
388C4—4222no—
389C4—4221no—
390C4C22⋊C4×D21111yescharacteristic
391C4C2×(C2×D211)⋊C411yescharacteristic
392C4#2026994511yescharacteristic
393C4#2026994711yescharacteristic
394C2—21no—
395C2—21no—
396C2—42no—
397C2—8442no—
398C2—8441no—
399C2—4222no—
400C2#2721300111yescharacteristic
401C2#104×D21111yescharacteristic
402C2#2721316111yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#41×D211C2
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1789
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#20269905C2semidirect
3. C2×#20269904C2semidirect
4. C2×#20269903C2semidirect
5. #27213109C2semidirect
6. C4×(C2×D211)⋊C4C2semidirect
7. C2×#104×C211C2semidirect
8. C2×#16809963C4semidirect
12. C2×#16809961C4semidirect
17. C2×Q8×D211C4semidirect
19. C2×D4×D211C4semidirect
22. C2×#16809946C22semidirect
24. C2×C4×Dic211C22semidirect
32. C2×M16×C211C22semidirect
33. C2×C42×C211C22semidirect
111. Dic422C22⋊C4semidirect
114. D844C22⋊C4semidirect
144. Q8×C211C22⋊C4semidirect
148. D4×C211C22⋊C4semidirect
181. C2×D211#104semidirect
201. C211#1789semidirect
207. C2×#104D211semidirect
226. C2×M16C2×D211semidirect
229. C2×C42C2×D211semidirect
317. Q8(C2×D211)⋊C4semidirect
328. D4(C2×D211)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2112
C2×#202699052
C2×#202699042
C2×#202699032
#272131092
C4×(C2×D211)⋊C42
C2×#104×C2112
#1789211