Finite Group Databasebeta

← #34008211

Subgroups — #34008211

Order 18688.

506 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#22625847C211yesmaximal, characteristic
2#22625716C211yesmaximal, characteristic
3#22624809C214yesmaximal
4#22618328C211yesmaximal, characteristic
5C2×#14556658C2212yes—
6C2×#14556658C2212yes—
7C2×#14556646—21no—
8#17989453—21no—
9#17989453—21no—
10#17989453—21no—
11C2×#14556590—21no—
12C2×#14556590—21no—
13C2×#14556590C412yes—
14#17988457C411yescharacteristic
15#17988456—21no—
16#17988456—21no—
17#17988456C2212yes—
18C42×Dic73C2211yescharacteristic
19#384×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—22no—
23#14556658—24no—
24#14556658Q814yescomplemented
25#14556646—42no—
26C22×Dic146—22no—
27#14556590—44no—
28#14556590—44no—
29#14556590—42no—
30#14556590—42no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73C2×C412yes—
34C2×#11924007—22no—
35C2×#11924007—22no—
36C2×#11924007—22no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73C2311yescharacteristic
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C412yes—
46C2×C42×C73C2×C411yescharacteristic
47C2×#10596833—24no—
48C2×#10596833—42no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×Dic146—44no—
54C2×Dic146—44no—
55C2×Dic146—44no—
56C22×Dic73—22no—
57C22×Dic73C22⋊C411yescharacteristic
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C2×D411yescharacteristic
60C22×Dic73C2×D411yescharacteristic
61C4⋊Dic73—42no—
62C4⋊Dic73—42no—
63C4⋊Dic73—42no—
64#11924007—44no—
65#11924007—42no—
66#11924007—42no—
67#11924007—44no—
68C4×Dic73—22no—
69C4×Dic73—22no—
70C4×Dic73—22no—
71C4×Dic73—22no—
72C4×Dic73D812yes—
73C4×Dic73SD1612yes—
74C4×Dic73C2×Q812yes—
75C4×Dic73#4112yes—
76C22×C4×C73—22no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C4⋊C4×C73—42no—
81C4⋊C4×C73—42no—
82C4⋊C4×C73—42no—
83C42×C73—22no—
84C23×C73C2×C22⋊C411yescharacteristic
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—42no—
92C2×C4×C73—42no—
93C2×C4×C73—42no—
94C2×C4×C73#10112yes—
95C2×C4×C73#10212yes—
96C2×C4×C73C4×Q812yes—
97C2×Dic73—21no—
98C2×Dic73—21no—
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—42no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104C2×Dic73#10012yes—
105C2×Dic73#10212yes—
106C2×Dic73C22⋊Q812yes—
107C2×Dic73#12411yescharacteristic
108C2×Dic73C4⋊Q811yescharacteristic
109C2×Dic73C2×D811yescharacteristic
110C2×Dic73C2×SD1611yescharacteristic
111C2×Dic73AGL(1,ℤ/8)12yes—
112C2×Dic73#13712yes—
113C2×Dic73#13712yes—
114Dic146—88no—
115#10596833—44no—
116#10596833—82no—
117C22×C73#39111yescharacteristic
118C22×C73C2×#10011yescharacteristic
119C22×C73C2×#10111yescharacteristic
120C22×C73C2×#10211yescharacteristic
121C22×C73#41811yescharacteristic
122C22×C73#41911yescharacteristic
123C22×C73#41911yescharacteristic
124C4×C73—22no—
125C4×C73—22no—
126C4×C73—22no—
127C4×C73—82no—
128Dic73—22no—
129Dic73—22no—
130Dic73—82no—
131Dic73#47412yes—
132Dic73#47512yes—
133Dic73#48912yes—
134Dic73#50112yes—
135#9905—731nomaximal, Sylow
136C2×C73#154511yescharacteristic
137C2×C73#154611yescharacteristic
138C2×C73#155611yescharacteristic
139C2×C73#155811yescharacteristic
140C2×C73#198811yescharacteristic
141C2×C73#199311yescharacteristic
142C2×C73#199711yescharacteristic
143#2868—731no—
144#1831—731no—
145#1767—731no—
146#1296—734no—
147C73#990511yescharacteristic, complemented, Sylow
148C2×C4⋊Q8—732no—
149C2×C4⋊Q8—1461no—
150C2×C4⋊Q8—1461no—
151C2×C4⋊C8—732no—
152C2×C4⋊C8—732no—
153C2×#97—1461no—
154#384—732no—
155#384—731no—
156#384—1461no—
157#384—1461no—
158#384#6440811yescharacteristic
159C43—731no—
160#336—1461no—
161#336—1461no—
162#336—1461no—
163C22×Q8—1462no—
164C22×C8—1462no—
165C22×C8—1462no—
166C4⋊Q8—2922no—
167C4⋊Q8—2922no—
168C4⋊Q8—2924no—
169C4⋊Q8—2924no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—732no—
173C2×C4⋊C4—1462no—
174C2×C4⋊C4—1462no—
175C2×C4⋊C4—1462no—
176C2×C4⋊C4—1461no—
177C2×C4⋊C4—1461no—
178C2×C4⋊C4C2×#6440812yes—
179C2×C42—731no—
180C2×C42—731no—
181C2×C42—731no—
182C2×C42—731no—
183C2×C42—731no—
184C2×C42—731no—
185C2×C42C2×#6440811yescharacteristic
186C4⋊C8—734no—
187C4⋊C8—1464no—
188C4⋊C8—1462no—
189#97—2922no—
190C2×Q8—2924no—
191C2×Q8—2924no—
192C2×Q8—2924no—
193C22×C4—22no—
194C22×C4—731no—
195C22×C4—731no—
196C22×C4—731no—
197C22×C4—731no—
198C22×C4—1462no—
199C22×C4C22⋊#6440811yescharacteristic
200C22×C4C22⋊#6440811yescharacteristic
201C22×C4C22×#6440811yescharacteristic
202C2×C8—2922no—
203C2×C8—2922no—
204C2×C8—2922no—
205C2×C8—2922no—
206C2×C8—2922no—
207C2×C8—1464no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—2922no—
212C4⋊C4—2922no—
213C4⋊C4—2924no—
214C4⋊C4—2922no—
215C4⋊C4—2922no—
216C4⋊C4—2924no—
217C4⋊C4—2922no—
218C42—22no—
219C42—732no—
220C42—732no—
221C42—732no—
222C42—732no—
223C42—1462no—
224C42—1462no—
225C42—1462no—
226C42—1462no—
227C23C2×C22⋊#6440811yescharacteristic
228Q8—5848no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—2922no—
239C2×C4—2922no—
240C2×C4—2922no—
241C2×C4—731no—
242C2×C4—732no—
243C2×C4—731no—
244C2×C4—731no—
245C2×C4—731no—
246C2×C4—732no—
247C2×C4—732no—
248C2×C4—732no—
249C2×C4—732no—
250C2×C4—732no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4—1461no—
254C2×C4—1461no—
255C2×C4#1455665212yes—
256C2×C4#1455665312yes—
257C2×C4Q8×#6440812yes—
258C8—2924no—
259C8—5842no—
260C22(C2×#64408)⋊Q811yescharacteristic
261C22D73⋊#10011yescharacteristic
262C22C2×#1455665211yescharacteristic
263C22C2×#1455665311yescharacteristic
264C22#1798957911yescharacteristic
265C22#1798958011yescharacteristic
266C22#1798958011yescharacteristic
267C4—22no—
268C4—22no—
269C4—22no—
270C4—82no—
271C4—5842no—
272C4—732no—
273C4—732no—
274C4—732no—
275C4—732no—
276C4—1462no—
277C4—1462no—
278C2#2262543911yescharacteristic
279C2#2262544011yescharacteristic
280C2#2262546011yescharacteristic
281C2#2262546511yescharacteristic
282C2#2262611011yescharacteristic
283C2#2262611511yescharacteristic
284C2#2262612411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#9905
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
24. #14556658Q8semidirect
147. C73#9905semidirect

Maximal subgroups

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

subgroupindex
#226258472
#226257162
#226248092
#226183282
#990573