Finite Group Databasebeta

← #33071262

Subgroups — #33071262

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620869C211yesmaximal, characteristic
2C2×#17988787C211yesmaximal, characteristic
3#22619012C211yesmaximal, characteristic, complemented
4#22609349C214yesmaximal, complemented
5#22609281C214yesmaximal, complemented
6#22608703C211yesmaximal, characteristic, complemented
7#22608600C211yesmaximal, characteristic
8#22608588C211yesmaximal, characteristic, complemented
9C73×#3078C211yesmaximal, characteristic, complemented
10C22×#11924015—21no—
11C2×#14556588C2211yescharacteristic
12#17988961C2212yes—
13#17988935C2212yes—
14#17988787—22no—
15#17988787—22no—
16#17988787C2214yes—
17#17988737—22no—
18Q8×#10596831—22no—
19Q8×#10596831C2214yescomplemented
20#17988640—22no—
21#17988640C2211yescharacteristic
22#17988640C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556511—21no—
25#17988337C2214yescomplemented
26#17988336C2212yescomplemented
27#17988333C2212yescomplemented
28#17988330C2214yes—
29#533×C73C2211yescharacteristic
30C2×C4×Q8×C73C2211yescharacteristic
31#479×C73—22no—
32#477×C73C2212yescomplemented
33#475×C73C2212yescomplemented
34#428×C73—21no—
35#423×C73C2211yescharacteristic
36#419×C73—21no—
37C22×D292—21no—
38C22×D292—21no—
39C22×D292—21no—
40C2×#11924015—24no—
41C2×#11924015—24no—
42C2×#11924015—44no—
43C22×#10596831—21no—
44#14556588—21no—
45#14556588—21no—
46#14556588—21no—
47#14556588C2312yes—
48C4⋊#10596831—24no—
49C4⋊#10596831—21no—
50C4⋊#10596831D412yes—
51C4⋊#10596831C2312yes—
52C4⋊#10596831C2312yes—
53#14556511—24no—
54#14556511—24no—
55C22⋊#10596831—21no—
56C22⋊#10596831D412yes—
57C4×#10596831—24no—
58C4×#10596831—21no—
59C4×#10596831C2312yes—
60C22×Q8×C73D411yescharacteristic
61C2×M16×C73—21no—
62C4⋊Q8×C73C2311yescharacteristic
63C4⋊Q8×C73C2311yescharacteristic
64#125×C73—21no—
65C22⋊Q8×C73—21no—
66C22⋊Q8×C73—21no—
67C4×Q8×C73—21no—
68C4×Q8×C73—21no—
69C4×Q8×C73—21no—
70C4×Q8×C73—21no—
71C4×Q8×C73D412yes—
72C4×Q8×C73C2312yes—
73#117×C73—21no—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73D411yescharacteristic
76C2×C42×C73—21no—
77C2×C42×C73C2311yescharacteristic
78#107×C73—22no—
79#106×C73—22no—
80C4⋊C8×C73—21no—
81C4⋊C8×C73C2312yes—
82#103×C73—22no—
83#103×C73—22no—
84C23×D73—41no—
85C2×D292—22no—
86C2×D292—22no—
87C2×D292—22no—
88C2×D292—42no—
89C2×D292—42no—
90C2×D292—41no—
91C2×D292—41no—
92C2×D292—41no—
93C2×D292—82no—
94C2×D292—82no—
95#11924015—48no—
96#11924015—84no—
97C2×#10596831—22no—
98C2×#10596831—22no—
99C2×#10596831—42no—
100C2×#10596831C2×D414yes—
101C2×Q8×C73—21no—
102C2×Q8×C73—22no—
103C2×Q8×C73—21no—
104C2×Q8×C73—21no—
105C2×Q8×C73SD1614yescomplemented
106C2×Q8×C73C2×D412yes—
107C22×C4×C73—21no—
108C22×C4×C73—21no—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73#4111yescharacteristic
112M16×C73—42no—
113C2×C8×C73—22no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—22no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73C2×D412yes—
124C22⋊C4×C73—41no—
125C42×C73—22no—
126C42×C73—22no—
127C42×C73—41no—
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73C2411yescharacteristic
132C23×C73#12111yescharacteristic
133C22×D73—42no—
134C22×D73—82no—
135C22×D73—82no—
136C22×D73—82no—
137Q8×C73—22no—
138Q8×C73—24no—
139Q8×C73—42no—
140Q8×C73—81no—
141Q8×C73C2×SD1614yes—
142C2×C4×C73—22no—
143C2×C4×C73—24no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—81no—
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12111yescharacteristic
153C2×C4×C73C2×SD1612yes—
154C2×C4×C73#13512yes—
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C22×D411yescharacteristic
157C2×C4×C73C2×#4111yescharacteristic
158D292—42no—
159D292—42no—
160D292—44no—
161D292—82no—
162D292—82no—
163D292—82no—
164D292—82no—
165C8×C73—42no—
166#10596831—24no—
167#10596831—42no—
168#10596831—42no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73#46512yes—
172C22×C73C2×#12111yescharacteristic
173C2×D73—82no—
174C2×D73—82no—
175C2×D73—82no—
176C2×D73—164no—
177C4×C73—21no—
178C4×C73—24no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—42no—
182C4×C73—81no—
183C4×C73—81no—
184C4×C73#46512yes—
185C4×C73C22×SD1611yescharacteristic
186C4×C73C2×#13511yescharacteristic
187C4×C73#57712yes—
188#19396—731nomaximal, Sylow
189C2×C73—22no—
190C2×C73#304311yescharacteristic
191C2×C73C2×#46511yescharacteristic
192C2×C73#306811yescharacteristic
193D73—162no—
194#3208—731no—
195#3078D7311yescharacteristic, complemented
196C2×#460—731no—
197#2982—731no—
198#1678—734no—
199#1666—734no—
200#1570—731no—
201#1545—731no—
202#1542—731no—
203C73#1939611yescharacteristic, complemented, Sylow
204C22×SD16—1461no—
205#533C2×D7311yescharacteristic
206C2×#127—731no—
207C2×C4×Q8C2×D7311yescharacteristic
208#492—732no—
209#486—732no—
210#479—22no—
211#477C2×D7312yescomplemented
212#475C2×D7312yescomplemented
213#460—734no—
214#460—1462no—
215#460—1462no—
216#450—1462no—
217C8×Q8—734no—
218C8×Q8—1462no—
219#431—731no—
220#431—731no—
221#431—1462no—
222#428—21no—
223#423C2×D7311yescharacteristic
224C2×C4⋊C8—731no—
225#419—21no—
226C2×#102—1461no—
227#334—734no—
228#333—732no—
229#332—732no—
230#331—734no—
231C22×Q8C22⋊D7311yescharacteristic
232C22×D4—1461no—
233C22×D4—1461no—
234C22×D4—1461no—
235C2×SD16—2924no—
236C2×SD16—1464no—
237C2×SD16—1464no—
238C2×M16—21no—
239C22×C8—1461no—
240C4⋊Q8C22×D7311yescharacteristic
241C4⋊Q8C22×D7311yescharacteristic
242#127—732no—
243#127—1461no—
244#127—1461no—
245#127—1461no—
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D7312yes—
254C4×Q8C22×D7312yes—
255#117—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D7311yescharacteristic
258C2×C42—21no—
259C2×C42C22×D7311yescharacteristic
260#107—22no—
261#106—22no—
262C4⋊C8—21no—
263C4⋊C8—732no—
264C4⋊C8—732no—
265C4⋊C8—732no—
266C4⋊C8—1464no—
267C4⋊C8—1461no—
268C4⋊C8C22×D7312yes—
269#103—22no—
270#103—22no—
271#102—1464no—
272#102—1464no—
273C22⋊C8—732no—
274C22⋊C8—1461no—
275C4×C8—732no—
276C4×C8—1464no—
277C4×C8—1461no—
278C24—2921no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8#1192401514yescomplemented
284C2×Q8C2×C22⋊D7312yes—
285C2×D4—2921no—
286C2×D4—2921no—
287C2×D4—2921no—
288C2×D4—2922no—
289C2×D4—2922no—
290C2×D4—5842no—
291C2×D4—5842no—
292C2×D4—1462no—
293C2×D4—1462no—
294C2×D4—1462no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4D4×D7311yescharacteristic
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4#1192402711yescharacteristic
300SD16—2928no—
301SD16—5844no—
302M16—42no—
303C2×C8—22no—
304C2×C8—2922no—
305C2×C8—734no—
306C2×C8—1462no—
307C2×C8—1462no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C2×C22⋊D7312yes—
318C22⋊C4—41no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42D4×D7311yescharacteristic
323C42C2×C22⋊D7311yescharacteristic
324C42#1192402711yescharacteristic
325C42C23×D7311yescharacteristic
326C23—2922no—
327C23—5842no—
328C23—5842no—
329C23—5842no—
330C23#1455656311yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×#1192401514yes—
336D4—2924no—
337D4—2922no—
338D4—2922no—
339D4—5842no—
340D4—5842no—
341D4—5842no—
342D4—5842no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#1455656311yescharacteristic
352C2×C4#1455656311yescharacteristic
353C2×C4#1455656311yescharacteristic
354C2×C4C2×#1192401512yes—
355C2×C4#1455661112yes—
356C2×C4C2×D4×D7311yescharacteristic
357C2×C4C22×C22⋊D7311yescharacteristic
358C2×C4C2×#1192402711yescharacteristic
359C8—42no—
360C8—2922no—
361C8—2922no—
362C8—1464no—
363C22—21no—
364C22—21no—
365C22—5842no—
366C22—5842no—
367C22—5842no—
368C22—11684no—
369C22#1798881812yes—
370C22C2×#1455656311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1798881812yes—
379C4C22×#1192401511yescharacteristic
380C4C2×#1455661111yescharacteristic
381C4#1798939612yes—
382C2—22no—
383C2—11682no—
384C2#2261945411yescharacteristic
385C2C2×#1798881811yescharacteristic
386C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3078C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#3078D73
hypocenterC73#19396
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #22619012C2semidirect
4. #22609349C2semidirect
5. #22609281C2semidirect
6. #22608703C2semidirect
8. #22608588C2semidirect
9. C73×#3078C2semidirect
19. Q8×#10596831C22semidirect
25. #17988337C22semidirect
26. #17988336C22semidirect
27. #17988333C22semidirect
32. #477×C73C22semidirect
33. #475×C73C22semidirect
105. C2×Q8×C73SD16semidirect
195. #3078D73semidirect
203. C73#19396semidirect
211. #477C2×D73semidirect
212. #475C2×D73semidirect
283. C2×Q8#11924015semidirect

Maximal subgroups

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

subgroupindex
#226208692
C2×#179887872
#226190122
#226093492
#226092812
#226087032
#226086002
#226085882
C73×#30782
#1939673