Finite Group Databasebeta

← #34030394

Subgroups — #34030394

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626317C212yesmaximal, complemented
2#22626316C214yesmaximal, complemented
3#22626315C212yesmaximal, complemented
4#22626291C211yesmaximal, characteristic, complemented
5#22626290C211yesmaximal, characteristic, complemented
6#22626160C212yesmaximal, complemented
7#22626159C212yesmaximal, complemented
8C8×D4×D73C211yesmaximal, characteristic
9C2×#14556666C2214yescomplemented
10C2×#14556665C2214yescomplemented
11#17989508C2212yescomplemented
12C4⋊#11924031C2212yescomplemented
13#17989482C2211yescharacteristic
14#17989482C2212yes—
15#17989482C2211yescharacteristic
16#17989482C2214yescomplemented
17C22⋊#11924031C2214yescomplemented
18#17989473C2212yescomplemented
19C4×#11924031C2212yescomplemented
20C22×C8×D73C2212yes—
21C4×D4×D73C411yescharacteristic
22C8×D292C411yescharacteristic
23D4×#10596831C2211yescharacteristic
24#17988652C2211yescharacteristic
25C8×C22⋊D73C412yes—
26#17988650C2212yes—
27C4⋊C8×D73C411yescharacteristic
28C22⋊C8×D73C412yes—
29C4×C8×D73C2211yescharacteristic
30C8×D4×C73C411yescharacteristic
31#14556666—44no—
32#14556665—44no—
33C2×#11924031C2312yes—
34C2×#11924031C2314yes—
35C2×#11924031C2312yes—
36C2×D4×D73C811yescharacteristic
37C22×C4×D73C2×C412yes—
38C22×#10596831C2312yes—
39C2×C8×D73—44no—
40C2×C8×D73C2×C412yes—
41C2×C8×D73C2×C411yescharacteristic
42C2×C8×D73C2×C412yes—
43C2×C8×D73C2312yes—
44C2×C8×D73C2311yescharacteristic
45C4×D292C2×C411yescharacteristic
46D4×Dic73C811yescharacteristic
47#14556553C811yescharacteristic
48C4×C22⋊D73C2×C412yes—
49#14556551C812yes—
50C4⋊C4×D73C811yescharacteristic
51C22⋊C4×D73C812yes—
52C42×D73C2×C411yescharacteristic
53C4⋊#10596831C2×C411yescharacteristic
54#14556520C2×C412yes—
55C22⋊#10596831C2×C412yes—
56#14556503C2×C412yes—
57#14556503C2×C412yes—
58C4×#10596831C2311yescharacteristic
59C8×Dic73C2×C411yescharacteristic
60C8×Dic73C2311yescharacteristic
61C22×C8×C73C2×C412yes—
62C4×D4×C73C2×C411yescharacteristic
63C4⋊C8×C73C2×C411yescharacteristic
64C22⋊C8×C73C2×C412yes—
65C4×C8×C73C2×C411yescharacteristic
66#11924031—22no—
67#11924031—24no—
68#11924031—22no—
69C23×D73C2×C812yes—
70C2×D292C2×C811yescharacteristic
71C2×C22⋊D73C2×C812yes—
72D4×D73—24no—
73C22×Dic73C2×C812yes—
74C2×C4×D73—42no—
75C2×C4×D73—42no—
76C2×C4×D73C2×C811yescharacteristic
77C2×C4×D73C2×C812yes—
78C2×C4×D73C2×C812yes—
79C2×C4×D73C2×C811yescharacteristic
80C2×C4×D73C22×C412yes—
81C2×C4×D73C22×C411yescharacteristic
82C2×#10596831—24no—
83C2×#10596831C22×C411yescharacteristic
84C2×#10596831C22×C412yes—
85C2×#10596831C2411yescharacteristic
86C8×D73—22no—
87C8×D73—24no—
88C8×D73—42no—
89C8×D73—42no—
90C8×D73—41no—
91C4⋊Dic73C2×C811yescharacteristic
92#11924007C2×C812yes—
93C22⋊Dic73C2×C812yes—
94#11924005C2×C812yes—
95#11924005C2×C812yes—
96C4×Dic73C2×C811yescharacteristic
97C4×Dic73C2×C811yescharacteristic
98C4×Dic73C22×C411yescharacteristic
99C2×D4×C73C2×C811yescharacteristic
100C22×C4×C73C22×C412yes—
101C2×C8×C73—42no—
102C2×C8×C73C22×C411yescharacteristic
103C2×C8×C73C22×C411yescharacteristic
104C2×C8×C73C22×C412yes—
105C4⋊C4×C73C2×C811yescharacteristic
106C22⋊C4×C73C2×C812yes—
107C42×C73C22×C411yescharacteristic
108C23×C73C22×C812yes—
109C22×D73—24no—
110C22×D73—42no—
111C22×D73—42no—
112C22×D73C22×C812yes—
113C22×D73C22×C811yescharacteristic
114D4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73C22×C811yescharacteristic
118C2×C4×C73C22×C812yes—
119C2×C4×C73C22×C811yescharacteristic
120C2×C4×C73C23×C411yescharacteristic
121C22⋊D73—24no—
122C2×Dic73—42no—
123C2×Dic73C22×C811yescharacteristic
124C2×Dic73C22×C811yescharacteristic
125C2×Dic73C22×C812yes—
126C2×Dic73C22×C811yescharacteristic
127D292—22no—
128C4×D73—21no—
129C4×D73—22no—
130C4×D73—21no—
131C4×D73—22no—
132C4×D73—42no—
133C4×D73—42no—
134C4×D73—41no—
135C4×D73M3214yescomplemented
136C8×C73—21no—
137C8×C73—22no—
138C8×C73—21no—
139#10596831—22no—
140#10596831—21no—
141#10596831H(5,𝔽2)12yes—
142C22×C73—22no—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73C23×C811yescharacteristic
146C2×D73—21no—
147C2×D73—21no—
148C2×D73—22no—
149C2×D73—22no—
150C2×D73—22no—
151C2×D73—48no—
152C2×D73C2×M3212yes—
153C4×C73—21no—
154C4×C73—22no—
155C4×C73C2×M3212yes—
156C4×C73#51811yescharacteristic
157C4×C73#56811yescharacteristic
158Dic73—21no—
159Dic73—21no—
160Dic73—22no—
161Dic73C2×M3212yes—
162#33489—731nomaximal, Sylow
163C2×C73—42no—
164C2×C73#297211yescharacteristic
165C2×C73C22×M3211yescharacteristic
166C2×C73#340711yescharacteristic
167D73—22no—
168D73—42no—
169C2×C8×D4—731no—
170#2169—732no—
171#2169—734no—
172#2169—732no—
173#2162—731no—
174#2162—731no—
175#2112—732no—
176#2112—732no—
177C73#3348911yescharacteristic, complemented, Sylow
178C23×C8—732no—
179C2×C4×D4—731no—
180C2×M32—734no—
181C2×M32—734no—
182C8×D4—732no—
183C8×D4—731no—
184C8×D4—731no—
185C8×D4—732no—
186C8×D4—731no—
187C8×D4#6440811yescharacteristic
188C2×C4⋊C8—731no—
189C2×C22⋊C8—732no—
190C2×C4×C8—731no—
191C4⋊C16—732no—
192C4⋊C16—732no—
193C22⋊C16—731no—
194C22⋊C16—732no—
195C22⋊C16—731no—
196C22⋊C16—734no—
197C22⋊C16—734no—
198C4×C16—732no—
199C4×C16—732no—
200C22×D4—731no—
201C23×C4—732no—
202C22×C8—2924no—
203C22×C8—732no—
204C22×C8—732no—
205C22×C8—731no—
206C22×C8—732no—
207C22×C8—731no—
208C22×C8—732no—
209C22×C8C2×#6440812yes—
210C4×D4—731no—
211C4×D4—732no—
212C4×D4—731no—
213C4×D4—732no—
214C4×D4—731no—
215C4×D4C2×#6440811yescharacteristic
216C2×C4⋊C4—731no—
217C2×C22⋊C4—732no—
218C2×C42—731no—
219M32—2924no—
220M32—2924no—
221C2×C16—732no—
222C2×C16—732no—
223C2×C16—734no—
224C4⋊C8—732no—
225C4⋊C8—731no—
226C4⋊C8C2×#6440811yescharacteristic
227C22⋊C8—732no—
228C22⋊C8—732no—
229C22⋊C8—732no—
230C22⋊C8C2×#6440812yes—
231C4×C8—731no—
232C4×C8—731no—
233C4×C8—731no—
234C4×C8C2×#6440811yescharacteristic
235C24—732no—
236C2×D4—732no—
237C2×D4—731no—
238C2×D4—1464no—
239C2×D4#1192403611yescharacteristic
240C22×C4—2922no—
241C22×C4—2922no—
242C22×C4—732no—
243C22×C4—731no—
244C22×C4—732no—
245C22×C4—732no—
246C22×C4—731no—
247C22×C4—732no—
248C22×C4—731no—
249C22×C4C22×#6440812yes—
250C2×C8—42no—
251C2×C8—2922no—
252C2×C8—2922no—
253C2×C8—2921no—
254C2×C8—731no—
255C2×C8—731no—
256C2×C8—732no—
257C2×C8—1464no—
258C2×C8—1462no—
259C2×C8—1464no—
260C2×C8C22×#6440811yescharacteristic
261C2×C8C22×#6440811yescharacteristic
262C2×C8C22×#6440812yes—
263C4⋊C4—732no—
264C4⋊C4—731no—
265C4⋊C4#1192403611yescharacteristic
266C22⋊C4—732no—
267C22⋊C4—732no—
268C22⋊C4—732no—
269C22⋊C4#1192403612yes—
270C42—731no—
271C42—731no—
272C42—731no—
273C42C22×#6440811yescharacteristic
274C16—1462no—
275C16—1464no—
276C16—1462no—
277C23—2922no—
278C23—2922no—
279C23—731no—
280C23—732no—
281C23—1464no—
282C23C2×#1192403612yes—
283D4—22no—
284D4—1464no—
285D4—1462no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4—2921no—
291C2×C4—2922no—
292C2×C4—731no—
293C2×C4—734no—
294C2×C4—731no—
295C2×C4—731no—
296C2×C4—732no—
297C2×C4—1461no—
298C2×C4—1462no—
299C2×C4—1461no—
300C2×C4—1462no—
301C2×C4C2×#1192403611yescharacteristic
302C2×C4C2×#1192403612yes—
303C2×C4C2×#1192403611yescharacteristic
304C2×C4C23×#6440811yescharacteristic
305C8—21no—
306C8—22no—
307C8—21no—
308C8—732no—
309C8—1462no—
310C8—1461no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—2928no—
315C22—732no—
316C22—1462no—
317C22—1462no—
318C22—1462no—
319C22—1461no—
320C22—1461no—
321C22C22×#1192403611yescharacteristic
322C4—21no—
323C4—22no—
324C4—732no—
325C4—1461no—
326C4—1461no—
327C4—1462no—
328C4#1798964312yes—
329C4#1798965411yescharacteristic
330C4#1798966611yescharacteristic
331C2—42no—
332C2—2922no—
333C2—1462no—
334C2#2262677911yescharacteristic
335C2C2×#1798964311yescharacteristic
336C2#2262689911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#64408
commutator, socleC22×C73C23×C8
FittingC8×D4×C73C4
prime radicalC2×D4×D73C8
cosocleC2×#10596831C24
hypercenterC8×D4#64408
hypocenterC73#33489
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22626317C2semidirect
2. #22626316C2semidirect
3. #22626315C2semidirect
4. #22626291C2semidirect
5. #22626290C2semidirect
6. #22626160C2semidirect
7. #22626159C2semidirect
9. C2×#14556666C22semidirect
10. C2×#14556665C22semidirect
11. #17989508C22semidirect
12. C4⋊#11924031C22semidirect
16. #17989482C22semidirect
17. C22⋊#11924031C22semidirect
18. #17989473C22semidirect
19. C4×#11924031C22semidirect
135. C4×D73M32semidirect
177. C73#33489semidirect

Maximal subgroups

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

subgroupindex
#226263172
#226263162
#226263152
#226262912
#226262902
#226261602
#226261592
C8×D4×D732
#3348973