Finite Group Databasebeta

← #33242853

Subgroups — #33242853

Order 18688.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic292C211yesmaximal, characteristic, complemented
2#22612160C211yesmaximal, characteristic, complemented
3#22612128C212yesmaximal, complemented
4#22612070C211yesmaximal, characteristic, complemented
5#22612025C212yesmaximal, complemented
6#22611976C211yesmaximal, characteristic, complemented
7#22611847C212yesmaximal, complemented
8C4×#14556535C211yesmaximal, characteristic, complemented
9D4×#11924003C211yesmaximal, characteristic, complemented
10C2×#17988390C212yesmaximal, complemented
11C73×#3334C211yesmaximal, characteristic
12C22×Dic292C2212yes—
13D4×Dic146—21no—
14C2×#14556535C2211yescharacteristic
15C2×#14556535C2212yes—
16C22×#11924003C2212yes—
17#17988967C2211yescharacteristic
18#17988828C2212yes—
19#17988798—21no—
20#17988745—22no—
21C4×Dic292C2211yescharacteristic
22#17988414C2212yes—
23#17988414C2211yescharacteristic
24C4⋊#11924003C2211yescharacteristic
25#17988390C2212yes—
26#17988390C2212yes—
27#17988390C2211yescharacteristic
28#17988390C2212yes—
29#17988390C2212yes—
30#17988388C2211yescharacteristic
31C22⋊#11924003C2212yes—
32C4×#11924003C2211yescharacteristic
33#547×C73—21no—
34#500×C73C2211yescharacteristic
35#480×C73—22no—
36#474×C73—21no—
37#466×C73C2212yes—
38C4×D8×C73C2211yescharacteristic
39C8×D4×C73C2211yescharacteristic
40C2×#107×C73C2212yes—
41C22×Dic146—22no—
42C2×Dic292—24no—
43C2×Dic292—22no—
44C2×Dic292—21no—
45C2×Dic292—42no—
46C2×Dic292C2312yes—
47C2×Dic292C2311yescharacteristic
48#14556591—21no—
49#14556590—41no—
50#14556570—22no—
51#14556568—42no—
52C4×Dic146—21no—
53D4×Dic73—41no—
54#14556535—22no—
55#14556535—42no—
56C2×#11924003—42no—
57C2×#11924003C2311yescharacteristic
58C2×#11924003C2311yescharacteristic
59C2×#11924003C2312yes—
60#14556525C2311yescharacteristic
61#14556516—21no—
62#14556516—22no—
63C2×#41×C73—41no—
64C2×D8×C73C2311yescharacteristic
65C22×C8×C73C2312yes—
66C4⋊Q8×C73—21no—
67#123×C73—42no—
68C22⋊Q8×C73—41no—
69#121×C73—22no—
70C4×D4×C73—21no—
71C4×D4×C73D411yescharacteristic
72C2×C4⋊C4×C73—22no—
73#107×C73—22no—
74#107×C73—21no—
75#107×C73C2311yescharacteristic
76#107×C73C2312yes—
77C4⋊C8×C73D411yescharacteristic
78#102×C73—21no—
79#102×C73—22no—
80C22⋊C8×C73D412yes—
81C4×C8×C73C2311yescharacteristic
82C2×Dic146—21no—
83C2×Dic146—22no—
84C2×Dic146—44no—
85C2×Dic146—42no—
86C2×Dic146—82no—
87C22×Dic73—42no—
88Dic292—22no—
89Dic292—21no—
90Dic292—22no—
91Dic292—44no—
92Dic292—81no—
93Dic292#4112yescomplemented
94C4⋊Dic73—21no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97#11924007—41no—
98#11924007—42no—
99C22⋊Dic73—42no—
100C4×Dic73—41no—
101#11924003—21no—
102#11924003—21no—
103#11924003—22no—
104#41×C73—84no—
105C2×Q8×C73—41no—
106C2×D4×C73—21no—
107C2×D4×C73—42no—
108C2×D4×C73D811yescharacteristic
109C22×C4×C73—42no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D412yes—
112D8×C73—21no—
113D8×C73—21no—
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73C2×D412yes—
118C2×C8×C73C2411yescharacteristic
119C4⋊C4×C73—21no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—42no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73D811yescharacteristic
124C22⋊C4×C73—41no—
125C22⋊C4×C73—42no—
126C22⋊C4×C73D812yes—
127C42×C73C2×D411yescharacteristic
128C23×C73—41no—
129C23×C73C2×D812yes—
130Q8×C73—82no—
131D4×C73—41no—
132D4×C73—41no—
133D4×C73—84no—
134D4×C73SD3214yescomplemented
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—84no—
140C2×C4×C73—82no—
141C2×C4×C73C2×D811yescharacteristic
142C2×C4×C73C2×D812yes—
143C2×C4×C73C2×D811yescharacteristic
144C2×C4×C73C22×D411yescharacteristic
145C2×Dic73—41no—
146C2×Dic73—41no—
147C2×Dic73—42no—
148C2×Dic73—84no—
149Dic146—22no—
150Dic146—41no—
151Dic146—42no—
152Dic146—42no—
153Dic146—84no—
154C8×C73—22no—
155C8×C73—21no—
156C8×C73C2×#4111yescharacteristic
157C8×C73H(5,𝔽2)11yescharacteristic
158C22×C73—22no—
159C22×C73—22no—
160C22×C73—81no—
161C22×C73—81no—
162C22×C73—81no—
163C22×C73C2×SD3214yes—
164C22×C73C22×D811yescharacteristic
165C4×C73—21no—
166C4×C73—22no—
167C4×C73—81no—
168C4×C73—82no—
169C4×C73C2×SD3212yes—
170C4×C73#54611yescharacteristic
171C4×C73#57711yescharacteristic
172Dic73—42no—
173Dic73—81no—
174Dic73—82no—
175#33865—731nomaximal, Sylow
176C2×C73—24no—
177C2×C73—161no—
178C2×C73#329411yescharacteristic
179C2×C73C22×SD3211yescharacteristic
180C2×C73#341711yescharacteristic
181#3334D7311yescharacteristic
182D4×Q16—731no—
183#2241—731no—
184#2235—732no—
185#2225—731no—
186#2217—732no—
187#2208—731no—
188#2185—732no—
189C4×SD32—731no—
190D4×C16—731no—
191C2×#358—732no—
192C73#3386511yescharacteristic, complemented, Sylow
193C22×Q16—732no—
194D4×Q8—1461no—
195#547—21no—
196C2×SD32—732no—
197C2×SD32—731no—
198C22×C16—732no—
199#500C2×D7311yescharacteristic
200#494—731no—
201#480—22no—
202#474—21no—
203#467—732no—
204#466C2×D7312yes—
205#462—1461no—
206#451—1462no—
207C4×Q16—731no—
208C4×D8C2×D7311yescharacteristic
209C8×D4C2×D7311yescharacteristic
210C2×#107C2×D7312yes—
211#367—731no—
212#367—732no—
213C4⋊C16—731no—
214#358—731no—
215#358—732no—
216#358—732no—
217#358—732no—
218#358—732no—
219#357—731no—
220C22⋊C16—732no—
221C4×C16—731no—
222C2×#41—41no—
223C22×Q8—1462no—
224C2×Q16—2922no—
225C2×Q16—731no—
226C2×Q16—732no—
227C2×Q16—1464no—
228C2×Q16—1462no—
229C2×Q16—1461no—
230C2×D8C22×D7311yescharacteristic
231C22×C8C22×D7312yes—
232C4⋊Q8—21no—
233C4⋊Q8—2921no—
234C4⋊Q8—1461no—
235#123—42no—
236C22⋊Q8—41no—
237C22⋊Q8—2922no—
238C22⋊Q8—1462no—
239#121—22no—
240C4×Q8—1461no—
241C4×D4—21no—
242C4×D4—2921no—
243C4×D4C22⋊D7311yescharacteristic
244C2×C4⋊C4—22no—
245SD32—2922no—
246SD32—1462no—
247C2×C16—2922no—
248C2×C16—731no—
249C2×C16—731no—
250C2×C16—732no—
251#107—22no—
252#107—21no—
253#107—731no—
254#107C22×D7311yescharacteristic
255#107C22×D7312yes—
256C4⋊C8C22⋊D7311yescharacteristic
257#103—1461no—
258#103—1462no—
259#102—21no—
260#102—22no—
261C22⋊C8C22⋊D7312yes—
262C4×C8C22×D7311yescharacteristic
263#41—84no—
264C2×Q8—41no—
265C2×Q8—2924no—
266C2×Q8—2922no—
267C2×Q8—5842no—
268C2×Q8—1462no—
269C2×Q8—1461no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4#1192401311yescharacteristic
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—2922no—
276C22×C4C2×C22⋊D7312yes—
277Q16—2924no—
278Q16—5841no—
279Q16—732no—
280Q16—1462no—
281Q16—1462no—
282Q16—1461no—
283D8—21no—
284D8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8C2×C22⋊D7311yescharacteristic
288C2×C8C2×C22⋊D7312yes—
289C2×C8C23×D7311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—2921no—
295C4⋊C4—2922no—
296C4⋊C4—2921no—
297C4⋊C4—2921no—
298C4⋊C4—1461no—
299C4⋊C4#1192401311yescharacteristic
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—2922no—
303C22⋊C4#1192401312yes—
304C42—2921no—
305C42C2×C22⋊D7311yescharacteristic
306C16—1461no—
307C16—1461no—
308C16—1462no—
309C23—41no—
310C23C2×#1192401312yes—
311Q8—82no—
312Q8—2921no—
313Q8—2922no—
314Q8—2922no—
315Q8—5844no—
316Q8—1462no—
317D4—41no—
318D4—41no—
319D4—84no—
320D4#1455653514yescomplemented
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—2922no—
330C2×C4—5844no—
331C2×C4C2×#1192401311yescharacteristic
332C2×C4C2×#1192401312yes—
333C2×C4C2×#1192401311yescharacteristic
334C2×C4C22×C22⋊D7311yescharacteristic
335C8—21no—
336C8—22no—
337C8#41×D7311yescharacteristic
338C8#1455663711yescharacteristic
339C22—22no—
340C22—22no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22C2×#1455653514yes—
345C22C22×#1192401311yescharacteristic
346C4—22no—
347C4—21no—
348C4—81no—
349C4—82no—
350C4—2922no—
351C4—5842no—
352C4—5841no—
353C4C2×#1455653512yes—
354C4#1798922511yescharacteristic
355C4#1798939511yescharacteristic
356C2—24no—
357C2—161no—
358C2#2262173811yescharacteristic
359C2C22×#1455653511yescharacteristic
360C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3334C2
socleC22×C73C22×D8
hypercenter#3334D73
hypocenterC73#33865
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. D4×Dic292C2semidirect
2. #22612160C2semidirect
3. #22612128C2semidirect
4. #22612070C2semidirect
5. #22612025C2semidirect
6. #22611976C2semidirect
7. #22611847C2semidirect
8. C4×#14556535C2semidirect
9. D4×#11924003C2semidirect
10. C2×#17988390C2semidirect
93. Dic292#41semidirect
134. D4×C73SD32semidirect
192. C73#33865semidirect
320. D4#14556535semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2922
#226121602
#226121282
#226120702
#226120252
#226119762
#226118472
C4×#145565352
D4×#119240032
C2×#179883902
C73×#33342
#3386573