Finite Group Databasebeta

← #34020395

Subgroups — #34020395

Order 18688.

950 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989606C211yesmaximal, characteristic
2C2×C4×#11924038C211yesmaximal, characteristic
3#22625727C212yesmaximal, complemented
4C4⋊C4×#10596833C212yesmaximal, complemented
5#22625672C214yesmaximal
6#22625650C212yesmaximal, complemented
7#22625588C212yesmaximal, complemented
8#22612324C211yesmaximal, characteristic
9C22×#11924038C2212yes—
10#17989606C418yescomplemented
11C2×C4⋊#10596833C2211yescharacteristic
12C2×C4⋊#10596833C2211yescharacteristic
13C2×#14556648C2212yes—
14C4×#11924038C418yescomplemented
15C2×#14556646C2214yes—
16C2×#14556646C2211yescharacteristic
17C2×#14556646C2211yescharacteristic
18C2×C4×#10596833C2214yes—
19C2×C4×#10596833C2212yes—
20C2×C4×#10596833C2212yes—
21#17989453C2214yes—
22#17989453C2214yes—
23C2×C42×D73C2211yescharacteristic
24#17988454C412yes—
25#17988453C411yescharacteristic
26#17988449C412yes—
27#17988444C412yes—
28C4⋊C4×Dic73C2212yes—
29#17988434C2212yes—
30#17988432C2212yes—
31#382×C73C411yescharacteristic
32C2×#11924038—24no—
33C2×#11924038—24no—
34C2×#11924038C2×C418yes—
35C22×#10596833C2312yes—
36C22×#10596833C2312yes—
37C22×#10596833C2312yes—
38C22×#10596833C2312yes—
39C4⋊#10596833C2×C414yes—
40C4⋊#10596833C2×C414yes—
41#14556648C2×C418yes—
42#14556646—28no—
43#14556646C2×C414yes—
44#14556646C2×C414yes—
45C4×#10596833—28no—
46C4×#10596833C2×C418yes—
47C4×#10596833C2×C418yes—
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2311yescharacteristic
51C2×C4⋊Dic73C2×C412yes—
52C2×#11924005C2×C414yes—
53C2×C4×Dic73C2×C411yescharacteristic
54C2×C4×Dic73C2314yes—
55C2×C4×Dic73C2311yescharacteristic
56C2×C4×Dic73C2311yescharacteristic
57C42×D73C2×C414yes—
58C42×D73C2×C414yes—
59#14556498C2×C414yes—
60#14556498C2×C412yes—
61#14556498C2×C412yes—
62#14556498C2×C414yes—
63C2×C4⋊C4×C73C2×C412yes—
64C2×C42×C73C2×C411yescharacteristic
65#95×C73C2×C412yes—
66#95×C73C2×C412yes—
67#11924038—216no—
68C2×#10596833—22no—
69C2×#10596833—22no—
70C2×#10596833—24no—
71C2×#10596833—24no—
72C2×#10596833C22×C414yes—
73C2×#10596833C22×C414yes—
74C2×#10596833C22×C414yes—
75C2×#10596833C22×C414yes—
76C2×#10596833#4114yes—
77C2×#10596833#4114yes—
78C23×D73C22×C411yescharacteristic
79C22×Dic73C22×C414yes—
80C22×Dic73C22×C411yescharacteristic
81C22×Dic73C22×C411yescharacteristic
82C22×Dic73C2411yescharacteristic
83C2×C4×D73—22no—
84C2×C4×D73—22no—
85C2×C4×D73—22no—
86C2×C4×D73—22no—
87C2×C4×D73—42no—
88C2×C4×D73C4214yes—
89C2×C4×D73C4214yes—
90C2×C4×D73C22×C412yes—
91C2×C4×D73C22×C412yes—
92C4⋊Dic73—24no—
93#11924005—44no—
94C4×Dic73—24no—
95C4×Dic73—24no—
96C4×Dic73C4214yes—
97C4×Dic73C22×C412yes—
98C4×Dic73C22×C412yes—
99C4×Dic73C22×C412yes—
100C4×Dic73C22×C412yes—
101C22×C4×C73C22×C414yes—
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104C22×C4×C73C22×C411yescharacteristic
105C4⋊C4×C73—24no—
106C42×C73C4214yes—
107C23×C73C23×C411yescharacteristic
108C22×D73—22no—
109C22×D73—21no—
110C22×D73—21no—
111C22×D73—22no—
112C22×D73C2×C4212yes—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—24no—
116C2×C4×C73—24no—
117C2×C4×C73—24no—
118C2×C4×C73C2×C4212yes—
119C2×C4×C73C2×C4212yes—
120C2×C4×C73C2×C4212yes—
121C2×C4×C73#13112yes—
122C2×C4×C73#13112yes—
123C2×Dic73—21no—
124C2×Dic73—21no—
125C2×Dic73—22no—
126C2×Dic73—24no—
127C2×Dic73—24no—
128C2×Dic73—24no—
129C2×Dic73C2×C4212yes—
130C2×Dic73C2×C4212yes—
131C2×Dic73C23×C411yescharacteristic
132C2×Dic73C23×C411yescharacteristic
133C2×Dic73C2×#4112yes—
134C2×Dic73C2×#4112yes—
135C4×D73—28no—
136C4×D73—24no—
137C4×D73—22no—
138C4×D73—22no—
139C4×D73—44no—
140#10596833—24no—
141#10596833—24no—
142#10596833—24no—
143#10596833#11718yes—
144C22×C73C22×C4211yescharacteristic
145C22×C73C4×#4112yes—
146C22×C73C2×#13111yescharacteristic
147C22×C73C2×#13111yescharacteristic
148C22×C73#56812yes—
149C2×D73—22no—
150C2×D73—22no—
151C2×D73—22no—
152C2×D73—42no—
153C2×D73—42no—
154C4×C73—22no—
155C4×C73—22no—
156C4×C73—44no—
157C4×C73#40514yes—
158Dic73—22no—
159Dic73—22no—
160Dic73—44no—
161Dic73C2×#11712yes—
162Dic73C4×#4111yescharacteristic
163Dic73C4×#4111yescharacteristic
164#20482—731nomaximal, Sylow
165C2×C73C4×#11711yescharacteristic
166C2×C73C2×#40511yescharacteristic
167C2×C73#287311yescharacteristic
168C2×C73#296912yes—
169C2×C73#296912yes—
170D73—42no—
171C2×#431—731no—
172C2×C4×M16—731no—
173#2283—731no—
174#1770—732no—
175C8×C4⋊C4—732no—
176#1753—734no—
177#1746—732no—
178#1726—732no—
179C73#2048211yescharacteristic, complemented, Sylow
180C22×M16—732no—
181C22×C42—731no—
182#431—738no—
183C2×C4⋊C8—731no—
184C2×C4⋊C8—731no—
185C2×C22⋊C8—732no—
186C4×M16—738no—
187C2×#97—731no—
188C2×#97—731no—
189C2×#97—734no—
190C2×C4×C8—732no—
191C2×C4×C8—732no—
192C2×C4×C8—734no—
193#383—732no—
194#382—731no—
195#382#6440811yescharacteristic
196#381—732no—
197#380—732no—
198C4×C4⋊C4—732no—
199C4×C22⋊C4—732no—
200#376—732no—
201#336—734no—
202#336—734no—
203C23×C4—731no—
204C23×C4—731no—
205C23×C4—731no—
206C2×M16—738no—
207C2×M16—1464no—
208C2×M16—1464no—
209C22×C8—732no—
210C22×C8—732no—
211C22×C8—732no—
212C22×C8—732no—
213C2×C4⋊C4—732no—
214C2×C4⋊C4C2×#6440812yes—
215C2×C22⋊C4—734no—
216C2×C42—731no—
217C2×C42—731no—
218C2×C42—731no—
219C2×C42—734no—
220C2×C42—734no—
221C2×C42—734no—
222C2×C42C2×#6440811yescharacteristic
223C4⋊C8—734no—
224C4⋊C8—734no—
225C22⋊C8—738no—
226#97—734no—
227#97—734no—
228#97—1468no—
229C4×C8—738no—
230C4×C8—738no—
231C4×C8—1468no—
232#95—734no—
233#95—732no—
234#95—732no—
235#95—734no—
236#95C2×#6440812yes—
237#95C2×#6440812yes—
238C24—731no—
239C22×C4—2922no—
240C22×C4—734no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4—731no—
244C22×C4—734no—
245C22×C4—732no—
246C22×C4—732no—
247C22×C4—734no—
248C22×C4—1462no—
249C22×C4—1462no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4C22×#6440814yes—
253C22×C4C22×#6440811yescharacteristic
254C22×C4C22×#6440811yescharacteristic
255C22×C4C22×#6440811yescharacteristic
256M16—14616no—
257C2×C8—734no—
258C2×C8—734no—
259C2×C8—734no—
260C2×C8—734no—
261C2×C8—734no—
262C2×C8—734no—
263C2×C8—1464no—
264C2×C8—1464no—
265C2×C8—1462no—
266C2×C8—1462no—
267C4⋊C4—24no—
268C4⋊C4—1464no—
269C22⋊C4—2924no—
270C42—734no—
271C42—732no—
272C42—732no—
273C42—732no—
274C42—732no—
275C42—1464no—
276C42—1464no—
277C42C4×#6440814yes—
278C23—732no—
279C23—1462no—
280C23—1461no—
281C23—1461no—
282C23—1462no—
283C23C23×#6440811yescharacteristic
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—2924no—
290C2×C4—731no—
291C2×C4—731no—
292C2×C4—732no—
293C2×C4—732no—
294C2×C4—732no—
295C2×C4—732no—
296C2×C4—1464no—
297C2×C4—1462no—
298C2×C4—1462no—
299C2×C4—1464no—
300C2×C4—1464no—
301C2×C4—1464no—
302C2×C4—1461no—
303C2×C4—1461no—
304C2×C4—1462no—
305C2×C4—1468no—
306C2×C4C2×C4×#6440812yes—
307C2×C4C2×C4×#6440812yes—
308C2×C4C2×C4×#6440812yes—
309C2×C4#1455667812yes—
310C2×C4#1455667912yes—
311C8—738no—
312C8—1464no—
313C8—1464no—
314C8—1464no—
315C22—2922no—
316C22—2922no—
317C22—1462no—
318C22—1462no—
319C22—1462no—
320C22C22×C4×#6440811yescharacteristic
321C22#41×#6440812yes—
322C22C2×#1455667811yescharacteristic
323C22C2×#1455667911yescharacteristic
324C22#1798966612yes—
325C4—22no—
326C4—22no—
327C4—44no—
328C4—2924no—
329C4—732no—
330C4—731no—
331C4—731no—
332C4—1462no—
333C4—1462no—
334C4#1798955214yes—
335C2—2922no—
336C2#117×#6440811yescharacteristic
337C2C2×#1798955211yescharacteristic
338C2#2262649611yescharacteristic
339C2#2262676312yes—
340C2#2262676712yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutatorC22×C73C22×C42
Fitting#382×C73C4
socleC23×C73C23×C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#382#64408
hypocenterC73#20482
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #22625727C2semidirect
4. C4⋊C4×#10596833C2semidirect
6. #22625650C2semidirect
7. #22625588C2semidirect
10. #17989606C4semidirect
14. C4×#11924038C4semidirect
179. C73#20482semidirect

Maximal subgroups

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

subgroupindex
C2×#179896062
C2×C4×#119240382
#226257272
C4⋊C4×#105968332
#226256722
#226256502
#226255882
#226123242
#2048273