Finite Group Databasebeta

← #34008228

Subgroups — #34008228

Order 18688.

592 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 114 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#22625788C211yesmaximal, characteristic
2#22625107C211yesmaximal, characteristic
3#22624804C214yesmaximal
4#22618284C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×#14556648C2212yes—
7C2×(C2×#64408)⋊C4—21no—
8C2×(C2×#64408)⋊C4—21no—
9C2×(C2×#64408)⋊C4C2212yes—
10#17989453—21no—
11C2×#14556591C411yescharacteristic, complemented
12C2×#14556571—21no—
13C2×C42×D73C2211yescharacteristic
14#17988492—21no—
15#17988486—21no—
16#17988486C412yescomplemented
17#17988444—21no—
18#17988444C2212yes—
19#384×C73C411yescharacteristic, complemented
20C23×#64408—22no—
21C22×#10596833—22no—
22#14556658—42no—
23#14556648—24no—
24(C2×#64408)⋊C4—22no—
25(C2×#64408)⋊C4—42no—
26(C2×#64408)⋊C4—42no—
27(C2×#64408)⋊C4Q814yescomplemented
28C22×Dic146—21no—
29C22×C4×D73D411yescharacteristic
30C22×C4×D73D411yescharacteristic
31C22×C4×D73C2311yescharacteristic
32#14556591—42no—
33#14556571—44no—
34C2×C4⋊Dic73—22no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73C2×C412yes—
37C2×#11924005—22no—
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C42×D73—42no—
42#14556498—22no—
43#14556498—22no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C412yes—
47C2×C42×C73C2×C411yescharacteristic
48C22×#64408—24no—
49C22×#64408—42no—
50C22×#64408—42no—
51C22×#64408—42no—
52C22×#64408—42no—
53C22×#64408—42no—
54C22×#64408—42no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C23×D73C2×D411yescharacteristic
59C2×Dic146—42no—
60C2×Dic146—42no—
61C2×Dic146—42no—
62C22×Dic73—22no—
63C22×Dic73C22⋊C411yescharacteristic
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73C2×D411yescharacteristic
66C2×C4×D73—22no—
67C2×C4×D73—22no—
68C2×C4×D73—41no—
69C2×C4×D73—41no—
70C2×C4×D73—41no—
71C2×C4×D73—41no—
72C2×C4×D73—41no—
73C2×C4×D73SD1612yes—
74C2×C4×D73Q1612yes—
75C2×C4×D73C2×Q812yes—
76C2×C4×D73#4112yes—
77C4⋊Dic73—44no—
78C4⋊Dic73—42no—
79C4⋊Dic73—42no—
80#11924005—44no—
81C4×Dic73—22no—
82C4×Dic73—41no—
83C4×Dic73—41no—
84C22×C4×C73—22no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C4⋊C4×C73—42no—
89C4⋊C4×C73—42no—
90C4⋊C4×C73—42no—
91C42×C73—22no—
92C23×C73C2×C22⋊C411yescharacteristic
93C22×D73—21no—
94C22×D73—21no—
95C22×D73—21no—
96C22×D73—21no—
97C22×D73C22⋊Q812yes—
98C22×D73C2×SD1611yescharacteristic
99C22×D73C2×Q1611yescharacteristic
100C22×D73AGL(1,ℤ/8)12yes—
101C2×#64408—44no—
102C2×#64408—44no—
103C2×#64408—44no—
104C2×#64408—82no—
105C2×#64408—82no—
106C2×#64408—82no—
107C2×#64408—82no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73#9912yes—
118C2×C4×C73#10312yes—
119C2×C4×C73C4×Q812yes—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—21no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—42no—
130C2×Dic73#10312yes—
131C2×Dic73#12411yescharacteristic
132C2×Dic73C4⋊Q811yescharacteristic
133C4×D73—42no—
134C4×D73—42no—
135C4×D73—42no—
136C4×D73—42no—
137C4×D73—44no—
138Dic146—84no—
139#10596833—82no—
140C22×C73#39111yescharacteristic
141C22×C73C2×#9911yescharacteristic
142C22×C73#41011yescharacteristic
143C22×C73C2×#10311yescharacteristic
144C22×C73#41811yescharacteristic
145C22×C73#42111yescharacteristic
146C22×C73#42111yescharacteristic
147C2×D73—22no—
148C2×D73—22no—
149C2×D73—22no—
150C2×D73—22no—
151C2×D73—21no—
152C2×D73—21no—
153C2×D73—21no—
154C2×D73—21no—
155C2×D73#47612yes—
156C2×D73#47712yes—
157#64408—84no—
158C4×C73—22no—
159C4×C73—22no—
160C4×C73—22no—
161C4×C73—82no—
162Dic73—21no—
163Dic73—21no—
164Dic73—22no—
165Dic73—41no—
166Dic73—82no—
167#9911—731nomaximal, Sylow
168C2×C73#151011yescharacteristic
169C2×C73#151411yescharacteristic
170C2×C73#151811yescharacteristic
171C2×C73#152011yescharacteristic
172C2×C73#198711yescharacteristic
173C2×C73#199511yescharacteristic
174C2×C73#199611yescharacteristic
175D73—22no—
176D73—22no—
177#2857—731no—
178#1797—731no—
179#1434—731no—
180#1294—734no—
181C73#991111yescharacteristic, complemented, Sylow
182C2×C4⋊Q8—731no—
183C2×C22⋊Q8—1461no—
184C22×C42—731no—
185C2×C4⋊C8—1461no—
186C2×C22⋊C8—732no—
187#391—1461no—
188#389—732no—
189#389—1461no—
190#384#6440811yescharacteristic, complemented
191#380—732no—
192#380—1461no—
193C2×#95—732no—
194C2×#95—1461no—
195C2×#95—1461no—
196#336—1461no—
197C22×Q8—1461no—
198C23×C4—731no—
199C23×C4—731no—
200C23×C4—731no—
201C23×C4—1462no—
202C22×C8—1462no—
203C4⋊Q8—2922no—
204C22⋊Q8—2924no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—732no—
208C2×C4⋊C4—1462no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4C2×#6440812yes—
211C2×C22⋊C4—1462no—
212C2×C42—2922no—
213C2×C42—731no—
214C2×C42—731no—
215C2×C42—731no—
216C2×C42C2×#6440811yescharacteristic
217C4⋊C8—2922no—
218C22⋊C8—1464no—
219#95—2922no—
220#95—2922no—
221#95—734no—
222#95—1462no—
223#95—1462no—
224#95—1462no—
225C24—731no—
226C2×Q8—2922no—
227C2×Q8—2922no—
228C2×Q8—2922no—
229C22×C4—22no—
230C22×C4—2921no—
231C22×C4—2921no—
232C22×C4—2921no—
233C22×C4—2921no—
234C22×C4—2921no—
235C22×C4—2922no—
236C22×C4—2922no—
237C22×C4—2922no—
238C22×C4—2922no—
239C22×C4—2922no—
240C22×C4—2922no—
241C22×C4—732no—
242C22×C4—732no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—731no—
246C22×C4—731no—
247C22×C4—731no—
248C22×C4—1462no—
249C22×C4—1464no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4C22⋊#6440811yescharacteristic
253C22×C4C22⋊#6440811yescharacteristic
254C22×C4C22×#6440811yescharacteristic
255C2×C8—2922no—
256C2×C8—2922no—
257C2×C8—2922no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—2922no—
262C4⋊C4—2922no—
263C4⋊C4—2924no—
264C22⋊C4—2924no—
265C42—22no—
266C42—2921no—
267C42—2921no—
268C42—1462no—
269C23—732no—
270C23—731no—
271C23—731no—
272C23—732no—
273C23—1461no—
274C23—1461no—
275C23—1461no—
276C23—1461no—
277C23C2×C22⋊#6440811yescharacteristic
278Q8—5844no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—2924no—
289C2×C4—2922no—
290C2×C4—2922no—
291C2×C4—2922no—
292C2×C4—2922no—
293C2×C4—2924no—
294C2×C4—2924no—
295C2×C4—2924no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—2922no—
299C2×C4—5842no—
300C2×C4—5842no—
301C2×C4—5842no—
302C2×C4—5842no—
303C2×C4—731no—
304C2×C4—731no—
305C2×C4—732no—
306C2×C4—1461no—
307C2×C4—1461no—
308C2×C4—1461no—
309C2×C4—1461no—
310C2×C4—1461no—
311C2×C4—1461no—
312C2×C4—1461no—
313C2×C4#1455665012yes—
314C2×C4#1455665412yes—
315C2×C4Q8×#6440812yes—
316C8—5842no—
317C22—732no—
318C22—732no—
319C22—1462no—
320C22—1462no—
321C22—1462no—
322C22—1462no—
323C22—1461no—
324C22—1461no—
325C22—1461no—
326C22—1461no—
327C22(C2×#64408)⋊Q811yescharacteristic
328C22C2×#1455665011yescharacteristic
329C22#1798956711yescharacteristic
330C22C2×#1455665411yescharacteristic
331C22#1798957911yescharacteristic
332C22#1798958511yescharacteristic
333C22#1798958611yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—82no—
338C4—2921no—
339C4—5844no—
340C4—5842no—
341C4—1461no—
342C4—1461no—
343C4—1462no—
344C2—1462no—
345C2—1462no—
346C2#2262533811yescharacteristic
347C2#2262535111yescharacteristic
348C2#2262536511yescharacteristic
349C2#2262537011yescharacteristic
350C2#2262610911yescharacteristic
351C2#2262611911yescharacteristic
352C2#2262612111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C2×D4
cosocleC22×C4×D73C23
hypercenter#384#64408
hypocenterC73#9911
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
11. C2×#14556591C4semidirect
16. #17988486C4semidirect
19. #384×C73C4semidirect
27. (C2×#64408)⋊C4Q8semidirect
181. C73#9911semidirect
190. #384#64408semidirect

Maximal subgroups

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

subgroupindex
#226257882
#226251072
#226248042
#226182842
#991173