Finite Group Databasebeta

← #34008150

Subgroups — #34008150

Order 18688.

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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#22625787C211yesmaximal, characteristic
2#22625657C211yesmaximal, characteristic
3#22624769C214yesmaximal
4C2×#17988492C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×C22⋊#10596833C2212yes—
7C2×C22⋊#10596833C2212yes—
8C2×#14556646—21no—
9#17989453—21no—
10#17989453—21no—
11C23×Dic146C411yescharacteristic
12C22×C4×Dic73C2211yescharacteristic
13#17988492—21no—
14#17988492—21no—
15#17988492—21no—
16#17988492—21no—
17C2×#14556498C412yes—
18C2×#14556498C2212yes—
19C22×C4⋊C4×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—42no—
23C22⋊#10596833—22no—
24C22⋊#10596833—24no—
25C22⋊#10596833Q814yescomplemented
26#14556646—42no—
27C22×Dic146—24no—
28C22×Dic146—21no—
29C22×Dic146—21no—
30C22×Dic146—42no—
31C22×Dic146—42no—
32C23×Dic73C2×C411yescharacteristic
33C23×Dic73C2×C412yes—
34C23×Dic73C2311yescharacteristic
35C2×C4×Dic73—42no—
36C2×C4×Dic73D411yescharacteristic
37C2×C4×Dic73D411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40#14556498—22no—
41#14556498—22no—
42#14556498—22no—
43#14556498—22no—
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73C2×C412yes—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—21no—
48C2×C4⋊C4×C73—42no—
49C2×#10596833—24no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C2×#10596833—42no—
55C2×Dic146—24no—
56C2×Dic146—24no—
57C2×Dic146—416no—
58C2×Dic146—42no—
59C2×Dic146—42no—
60C2×Dic146—42no—
61C2×Dic146—42no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—22no—
65C22×Dic73—22no—
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73—41no—
69C22×Dic73—42no—
70C22×Dic73—42no—
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C22⋊C411yescharacteristic
73C22×Dic73M1614yes—
74C22×Dic73C2×D411yescharacteristic
75C22×Dic73C2×D411yescharacteristic
76C22×Dic73C2×Q812yes—
77C22×Dic73#4112yes—
78C4×Dic73—22no—
79C4×Dic73—22no—
80C4×Dic73—41no—
81C4×Dic73—41no—
82C24×C73C22×C411yescharacteristic
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—24no—
88C22×C4×C73—42no—
89C22×C4×C73—42no—
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22⋊C411yescharacteristic
92C4⋊C4×C73—42no—
93C4⋊C4×C73—42no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73C2×C22⋊C411yescharacteristic
99C23×C73C4×Q812yes—
100C23×C73C2×M1612yes—
101C23×C73#13112yes—
102C2×C4×C73—28no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—84no—
111C2×C4×C73#9912yes—
112C2×C4×C73#10012yes—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—24no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—44no—
126C2×Dic73—44no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—44no—
130C2×Dic73#10112yes—
131C2×Dic73C22⋊Q812yes—
132C2×Dic73#12411yescharacteristic
133C2×Dic73C4⋊Q811yescharacteristic
134Dic146—48no—
135Dic146—48no—
136#10596833—44no—
137#10596833—82no—
138C22×C73—22no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—22no—
142C22×C73—22no—
143C22×C73—22no—
144C22×C73#39111yescharacteristic
145C22×C73#40711yescharacteristic
146C22×C73#40811yescharacteristic
147C22×C73C2×#9911yescharacteristic
148C22×C73#41011yescharacteristic
149C22×C73C2×#10011yescharacteristic
150C22×C73C2×#10111yescharacteristic
151C22×C73#44614yes—
152C4×C73—22no—
153C4×C73—22no—
154C4×C73—82no—
155Dic73—22no—
156Dic73—22no—
157Dic73—21no—
158Dic73—21no—
159Dic73—44no—
160Dic73—41no—
161#9882—731nomaximal, Sylow
162C2×C73—24no—
163C2×C73#146211yescharacteristic
164C2×C73#146711yescharacteristic
165C2×C73#146911yescharacteristic
166C2×C73#147211yescharacteristic
167C2×C73#198511yescharacteristic
168C2×C73#199611yescharacteristic
169C2×C73#199711yescharacteristic
170C2×#391—731no—
171#1797—731no—
172#1749—731no—
173#1280—734no—
174C73#988211yescharacteristic, complemented, Sylow
175C23×Q8—731no—
176C22×C4⋊C4#6440811yescharacteristic
177C22×C42—731no—
178C2×C4⋊C8—1461no—
179C2×C22⋊C8—732no—
180C2×C22⋊C8—732no—
181C2×#97—1461no—
182#391—1461no—
183#391—1461no—
184#391—1461no—
185#391—1461no—
186C2×#95—732no—
187C2×#95—732no—
188#336—1461no—
189#336—1461no—
190C22×Q8—2922no—
191C22×Q8—2922no—
192C22×Q8—1464no—
193C22×Q8—1461no—
194C22×Q8—1461no—
195C23×C4—731no—
196C23×C4—732no—
197C23×C4—731no—
198C23×C4C2×#6440812yes—
199C23×C4C2×#6440811yescharacteristic
200C22×C8—1462no—
201C22×C8—1462no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C42—2922no—
206C2×C42—731no—
207C2×C42—731no—
208C2×C42—731no—
209C2×C42—731no—
210C4⋊C8—2922no—
211C22⋊C8—734no—
212C22⋊C8—1464no—
213C22⋊C8—1462no—
214#97—2922no—
215#95—1462no—
216#95—1462no—
217#95—1462no—
218#95—1462no—
219C24C22×#6440811yescharacteristic
220C2×Q8—29216no—
221C2×Q8—2922no—
222C2×Q8—2922no—
223C2×Q8—2922no—
224C2×Q8—2922no—
225C2×Q8—1464no—
226C2×Q8—1464no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—2922no—
235C22×C4—2922no—
236C22×C4—2921no—
237C22×C4—2921no—
238C22×C4—2921no—
239C22×C4—732no—
240C22×C4—732no—
241C22×C4—734no—
242C22×C4—731no—
243C22×C4—731no—
244C22×C4—731no—
245C22×C4—731no—
246C22×C4—1462no—
247C22×C4—1462no—
248C22×C4—1462no—
249C22×C4—1462no—
250C22×C4C22⋊#6440811yescharacteristic
251C22×C4C22⋊#6440811yescharacteristic
252C2×C8—2922no—
253C2×C8—2922no—
254C2×C8—2922no—
255C2×C8—2922no—
256C2×C8—2922no—
257C2×C8—1464no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42—2921no—
261C42—2921no—
262C42—1462no—
263C42—1462no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C2×C22⋊#6440811yescharacteristic
269C23Q8×#6440812yes—
270C23D73⋊M1612yes—
271C23#1455667812yes—
272Q8—2928no—
273Q8—2928no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—2924no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2924no—
287C2×C4—2924no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4—731no—
291C2×C4—732no—
292C2×C4—731no—
293C2×C4—732no—
294C2×C4—1464no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1461no—
303C2×C4—1461no—
304C2×C4#1455665012yes—
305C2×C4#1455665112yes—
306C8—2924no—
307C8—5842no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22(C2×#64408)⋊Q811yescharacteristic
315C22#1798955911yescharacteristic
316C22#1798956111yescharacteristic
317C22C2×#1455665011yescharacteristic
318C22#1798956711yescharacteristic
319C22C2×#1455665111yescharacteristic
320C22D73⋊#10111yescharacteristic
321C22#1798963914yes—
322C4—22no—
323C4—22no—
324C4—82no—
325C4—2924no—
326C4—2921no—
327C4—1462no—
328C4—1461no—
329C4—1461no—
330C4—1462no—
331C2—24no—
332C2#2262517811yescharacteristic
333C2#2262519511yescharacteristic
334C2#2262520111yescharacteristic
335C2#2262520911yescharacteristic
336C2#2262610311yescharacteristic
337C2#2262612111yescharacteristic
338C2#2262612311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#9882
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
25. C22⋊#10596833Q8semidirect
174. C73#9882semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226256572
#226247692
C2×#179884922
#988273