Finite Group Databasebeta

← #36369556

Subgroups — #36369556

Order 22144.

402 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 80 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×D4×#10621924C211yesmaximal, characteristic
2C2×#19149742C211yesmaximal, characteristic, complemented
3C2×#19149741C211yesmaximal, characteristic
4(C2×#10621924)⋊Q8C211yesmaximal, characteristic, complemented
5#26012365C211yesmaximal, characteristic, complemented
6(C2×#10621924)⋊C8C211yesmaximal, characteristic, complemented
7C2×SD16×D173C211yesmaximal, characteristic
8C23×#10621924—21no—
9C2×D173⋊C8C2211yescharacteristic
10D4×#10621924—22no—
11D4×#10621924—22no—
12C2×C4⋊#10621924C2211yescharacteristic
13C2×C4⋊#10621924C2211yescharacteristic
14C2×C22⋊#10621924—21no—
15C2×C4×#10621924C2211yescharacteristic
16#19149742—22no—
17#19149741—22no—
18(C2×#10621924)⋊C4—21no—
19(C2×#10621924)⋊C4—21no—
20C2×Q8×D173C2211yescharacteristic
21C2×D4×D173C2211yescharacteristic
22C2×#14583815C411yescharacteristic, complemented
23C2×#14583814C411yescharacteristic, complemented
24C2×#14583813C411yescharacteristic, complemented
25SD16×D173—22no—
26SD16×D173—22no—
27C2×C8×D173C2211yescharacteristic
28C2×SD16×C173C411yescharacteristic, complemented
29C22×#10621924—24no—
30C22×#10621924—21no—
31C22×#10621924—21no—
32C22×#10621924—42no—
33C22×#10621924—42no—
34C22×#10621924D411yescharacteristic
35C22×#10621924D411yescharacteristic
36D173⋊C8—21no—
37D173⋊C8—21no—
38C2×#13063443—21no—
39C4⋊#10621924—21no—
40C4⋊#10621924—21no—
41C4⋊#10621924—21no—
42C4⋊#10621924D412yes—
43C22⋊#10621924—24no—
44C4×#10621924—21no—
45C4×#10621924—21no—
46C23×D173—21no—
47C2×Dic346—21no—
48C2×Dic346C2×C411yescharacteristic, complemented
49Q8×D173—21no—
50Q8×D173—21no—
51Q8×D173—21no—
52Q8×D173—21no—
53C2×D692C2×C411yescharacteristic
54C2×C22⋊D173—21no—
55D4×D173—21no—
56D4×D173—21no—
57D4×D173D412yes—
58D4×D173D412yes—
59C2×C4×D173—21no—
60C2×C4×D173C2311yescharacteristic
61#14583815—22no—
62#14583814—22no—
63#14583813—22no—
64C2×#13063436C2×C411yescharacteristic, complemented
65C8×D173—21no—
66C8×D173—21no—
67C2×Q8×C173C2×C411yescharacteristic, complemented
68C2×D4×C173C2×C411yescharacteristic
69SD16×C173—22no—
70C2×C8×C173C2×C411yescharacteristic, complemented
71C2×#10621924—21no—
72C2×#10621924—21no—
73C2×#10621924—21no—
74C2×#10621924—22no—
75C2×#10621924—21no—
76C2×#10621924—21no—
77C2×#10621924—48no—
78C2×#10621924—41no—
79C2×#10621924—41no—
80C2×#10621924—41no—
81C2×#10621924—41no—
82C2×#10621924—41no—
83C2×#10621924SD1612yescomplemented
84#13063443—41no—
85C22×D173—21no—
86C22×D173—22no—
87C22×D173—22no—
88C22×D173—41no—
89C22×D173—41no—
90C22×D173—41no—
91C22×D173—41no—
92C22×D173C2×D411yescharacteristic
93Dic346—21no—
94Dic346—21no—
95Dic346—41no—
96Dic346—41no—
97D692—21no—
98D692C22⋊C412yes—
99C22⋊D173—22no—
100C22⋊D173—22no—
101C2×Dic173—21no—
102C2×Dic173#4111yescharacteristic
103C4×D173—41no—
104C4×D173—41no—
105C4×D173C2×D411yescharacteristic
106C4×D173C2×D411yescharacteristic
107C4×D173C2×D411yescharacteristic
108C4×D173#4111yescharacteristic
109#13063436—21no—
110C23×C173—21no—
111Q8×C173—21no—
112Q8×C173—21no—
113D4×C173—21no—
114D4×C173C22⋊C412yes—
115C2×C4×C173—21no—
116C2×C4×C173C22×C411yescharacteristic
117C8×C173—21no—
118C22×C173—22no—
119C22×C173—41no—
120C22×C173—41no—
121C22×C173C4×D411yescharacteristic
122C2×D173—22no—
123C2×D173—41no—
124C2×D173—41no—
125C2×D173—42no—
126C2×D173—42no—
127C2×D173—42no—
128C2×D173—42no—
129C2×D173#12011yescharacteristic
130C2×D173#12111yescharacteristic
131C2×D173C2×SD1611yescharacteristic
132C2×D173#13511yescharacteristic
133C2×D173AGL(1,ℤ/8)11yescharacteristic
134C2×D173#13711yescharacteristic
135#10621924—22no—
136#10621924—42no—
137#10621924—41no—
138#10621924—81no—
139C4×C173—41no—
140C4×C173C2×C22⋊C411yescharacteristic
141C4×C173C4×D411yescharacteristic
142Dic173—41no—
143Dic173#12111yescharacteristic
144Dic173#12311yescharacteristic
145C2×C173—42no—
146C2×C173#38611yescharacteristic
147C2×C173C4×SD1611yescharacteristic
148C2×C173#44011yescharacteristic
149D173—42no—
150D173#45011yescharacteristic
151D173#45211yescharacteristic
152D173#46111yescharacteristic
153D173#46311yescharacteristic
154C173#187711yescharacteristic, complemented, Sylow
155#1877—1731nomaximal, Sylow
156C22×SD16—1731no—
157C2×C4×D4—1731no—
158C2×#103—1731no—
159C2×#102—1731no—
160#391—1731no—
161#340—1731no—
162#336—1731no—
163C22×Q8—1731no—
164C22×D4—1731no—
165C23×C4—3461no—
166C2×SD16—3462no—
167C2×SD16—3462no—
168C2×SD16—1731no—
169C2×SD16—1731no—
170C2×SD16—1731no—
171C2×SD16#1062192411yescharacteristic, complemented
172C22×C8—1731no—
173C22×C8—1731no—
174C4×D4—3462no—
175C4×D4—3462no—
176C2×C4⋊C4—1731no—
177C2×C4⋊C4—1731no—
178C2×C22⋊C4—3461no—
179C2×C42—1731no—
180#103—3462no—
181#102—3462no—
182#95—3461no—
183#95—3461no—
184C24—3461no—
185C2×Q8—3461no—
186C2×Q8—3461no—
187C2×Q8—3461no—
188C2×Q8—3461no—
189C2×Q8—3461no—
190C2×Q8—1731no—
191C2×Q8C2×#1062192411yescharacteristic, complemented
192C2×D4—3461no—
193C2×D4—3461no—
194C2×D4—3461no—
195C2×D4—1732no—
196C2×D4—1732no—
197C2×D4—1731no—
198C2×D4C2×#1062192411yescharacteristic
199C22×C4—3461no—
200C22×C4—3464no—
201C22×C4—3461no—
202C22×C4—3461no—
203C22×C4—1731no—
204C22×C4—1731no—
205C22×C4—1731no—
206C22×C4—6922no—
207C22×C4—6922no—
208SD16—22no—
209SD16—3462no—
210SD16—3462no—
211SD16—3462no—
212C2×C8—3461no—
213C2×C8—3461no—
214C2×C8—3461no—
215C2×C8—3461no—
216C2×C8—3461no—
217C2×C8—1731no—
218C2×C8C2×#1062192411yescharacteristic, complemented
219C4⋊C4—3461no—
220C4⋊C4—3461no—
221C4⋊C4—3461no—
222C4⋊C4—1732no—
223C22⋊C4—3464no—
224C42—3461no—
225C42—3461no—
226C23—21no—
227C23—3462no—
228C23—3462no—
229C23—3461no—
230C23—1731no—
231C23—6921no—
232C23—6921no—
233C23—6921no—
234C23—6921no—
235Q8—21no—
236Q8—21no—
237Q8—3461no—
238Q8—3461no—
239Q8—6921no—
240Q8—6921no—
241D4—21no—
242D4—3462no—
243D4—3462no—
244D4—3461no—
245D4—1732no—
246D4C22⋊#1062192412yes—
247C2×C4—21no—
248C2×C4—3461no—
249C2×C4—3461no—
250C2×C4—3461no—
251C2×C4—3462no—
252C2×C4—3461no—
253C2×C4—3461no—
254C2×C4—3461no—
255C2×C4—1732no—
256C2×C4—1731no—
257C2×C4—1731no—
258C2×C4—1731no—
259C2×C4—1731no—
260C2×C4—1731no—
261C2×C4—6921no—
262C2×C4—6921no—
263C2×C4—6928no—
264C2×C4—6921no—
265C2×C4—6921no—
266C2×C4—6921no—
267C2×C4—6921no—
268C2×C4—6921no—
269C2×C4C22×#1062192411yescharacteristic
270C8—21no—
271C8—3461no—
272C8—6921no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—3462no—
277C22—1731no—
278C22—1731no—
279C22—1731no—
280C22—1731no—
281C22—1731no—
282C22—1731no—
283C22—6922no—
284C22—6922no—
285C22—6922no—
286C22—6922no—
287C22—6921no—
288C22—6921no—
289C22D4×#1062192411yescharacteristic
290C4—41no—
291C4—3462no—
292C4—13841no—
293C4—1731no—
294C4—1731no—
295C4—6921no—
296C4—6922no—
297C4—6921no—
298C4C2×C22⋊#1062192411yescharacteristic
299C4D4×#1062192411yescharacteristic
300C2—42no—
301C2—1731no—
302C2—1731no—
303C2—1731no—
304C2—1731no—
305C2—6922no—
306C2#2601243911yescharacteristic
307C2SD16×#1062192411yescharacteristic
308C2#2601253011yescharacteristic
subgroupG/H
centerC22D4×#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×SD16×C173C4
socleC22×C173C4×D4
prime radicalC2×D4×D173C22
cosocleC2×C4×D173C23
hypercenterC2×SD16#10621924
hypocenterC173#1877
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#19149742C2semidirect
4. (C2×#10621924)⋊Q8C2semidirect
5. #26012365C2semidirect
6. (C2×#10621924)⋊C8C2semidirect
22. C2×#14583815C4semidirect
23. C2×#14583814C4semidirect
24. C2×#14583813C4semidirect
28. C2×SD16×C173C4semidirect
48. C2×Dic346C2×C4semidirect
64. C2×#13063436C2×C4semidirect
67. C2×Q8×C173C2×C4semidirect
70. C2×C8×C173C2×C4semidirect
83. C2×#10621924SD16semidirect
154. C173#1877semidirect
171. C2×SD16#10621924semidirect
191. C2×Q8C2×#10621924semidirect
218. C2×C8C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#106219242
C2×#191497422
C2×#191497412
(C2×#10621924)⋊Q82
#260123652
(C2×#10621924)⋊C82
C2×SD16×D1732
#1877173