Finite Group Databasebeta

← #33057499

Subgroups — #33057499

Order 18688.

786 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11924012C212yesmaximal, complemented
2#22608845C212yesmaximal, complemented
3#22608796C214yesmaximal, complemented
4#22608755C212yesmaximal, complemented
5#22608687C211yesmaximal, characteristic, complemented
6#22608356C212yesmaximal, complemented
7#22608220C211yesmaximal, characteristic, complemented
8C2×C4×D8×C73C211yesmaximal, characteristic
9C22×#11924012—22no—
10#17988665C2214yescomplemented
11#17988659—24no—
12D4×#10596831C2214yescomplemented
13#17988640C2211yescharacteristic
14#17988611C2212yes—
15#17988551—22no—
16C4×#11924012C2211yescharacteristic
17#17988338C2212yescomplemented
18#17988338C2212yescomplemented
19D4⋊#10596831C2214yescomplemented
20D4⋊#10596831C2214yescomplemented
21#17988309C2212yescomplemented
22#17988308C2212yescomplemented
23C22×D8×C73C411yescharacteristic
24C2×C4×D4×C73C2212yes—
25C4×D8×C73C412yes—
26C4×D8×C73C411yescharacteristic
27C4×D8×C73C411yescharacteristic
28C4×D8×C73C2211yescharacteristic
29C4×D8×C73C2212yes—
30C4×D8×C73C2211yescharacteristic
31C2×#107×C73C411yescharacteristic
32C2×#102×C73C412yes—
33C2×C4×C8×C73C2211yescharacteristic
34C2×#11924012—24no—
35C2×#11924012—22no—
36C2×#11924012—48no—
37C2×#11924012D412yes—
38C22×#10596831—24no—
39C4⋊#10596831—22no—
40C4⋊#10596831—21no—
41C4⋊#10596831C2314yes—
42C22⋊#10596831—24no—
43C22⋊#10596831—24no—
44C22⋊#10596831—21no—
45#14556502D412yes—
46C4×#10596831C2312yes—
47C4×#10596831C2312yes—
48C22×D4×C73C2×C412yes—
49C23×C4×C73—22no—
50C2×D8×C73—24no—
51C2×D8×C73C2×C412yes—
52C2×D8×C73C2×C411yescharacteristic
53C2×D8×C73C2×C411yescharacteristic
54C22×C8×C73C2×C411yescharacteristic
55C22×C8×C73C2×C411yescharacteristic
56C4×D4×C73—22no—
57C4×D4×C73—22no—
58C4×D4×C73C2×C412yes—
59C4×D4×C73C2×C412yes—
60C4×D4×C73C2312yes—
61C4×D4×C73C2312yes—
62C2×C4⋊C4×C73C2×C412yes—
63C2×C22⋊C4×C73—22no—
64C2×C42×C73C2311yescharacteristic
65#107×C73C2×C412yes—
66#107×C73C2×C411yescharacteristic
67#107×C73C2×C411yescharacteristic
68#102×C73C2×C412yes—
69#102×C73C2×C412yes—
70#102×C73C2×C412yes—
71#102×C73C2×C412yes—
72C4×C8×C73C2×C411yescharacteristic
73C4×C8×C73C2×C411yescharacteristic
74C4×C8×C73C2311yescharacteristic
75C4×C8×C73C2311yescharacteristic
76#11924012—24no—
77#11924012—48no—
78#11924012—44no—
79C2×#10596831—24no—
80C2×#10596831—22no—
81C2×#10596831—48no—
82C2×#10596831C2×D414yes—
83C24×C73—22no—
84C2×D4×C73—24no—
85C2×D4×C73—22no—
86C2×D4×C73—42no—
87C2×D4×C73C22×C412yes—
88C2×D4×C73C22×C412yes—
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—22no—
93C22×C4×C73—22no—
94C22×C4×C73—44no—
95C22×C4×C73—44no—
96C22×C4×C73C22×C411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73#4111yescharacteristic
99D8×C73—24no—
100D8×C73—22no—
101D8×C73M1614yescomplemented
102C2×C8×C73—22no—
103C2×C8×C73—41no—
104C2×C8×C73C22×C411yescharacteristic
105C2×C8×C73C22×C411yescharacteristic
106C2×C8×C73C22×C411yescharacteristic
107C2×C8×C73C22×C411yescharacteristic
108C4⋊C4×C73—22no—
109C4⋊C4×C73C22×C412yes—
110C4⋊C4×C73C22×C412yes—
111C22⋊C4×C73—22no—
112C22⋊C4×C73—22no—
113C22⋊C4×C73—22no—
114C22⋊C4×C73—22no—
115C42×C73C22×C411yescharacteristic
116C42×C73C2×D411yescharacteristic
117C42×C73#4111yescharacteristic
118C42×C73C2411yescharacteristic
119C23×C73—22no—
120C23×C73—22no—
121C23×C73—44no—
122C23×C73—44no—
123C23×C73—82no—
124C23×C73C4×D411yescharacteristic
125D4×C73—22no—
126D4×C73—22no—
127D4×C73—22no—
128D4×C73—44no—
129D4×C73C2×M1614yescomplemented
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—41no—
135C2×C4×C73—44no—
136C2×C4×C73—42no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73—44no—
141C2×C4×C73—44no—
142C2×C4×C73C4×D411yescharacteristic
143C2×C4×C73C4×D411yescharacteristic
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73AGL(1,ℤ/8)14yes—
146C2×C4×C73C23×C411yescharacteristic
147C2×C4×C73C22×D411yescharacteristic
148C2×C4×C73C2×#4111yescharacteristic
149C8×C73—21no—
150C8×C73—21no—
151C8×C73—21no—
152C8×C73C2×M1612yescomplemented
153#10596831—24no—
154#10596831—42no—
155#10596831—44no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—24no—
159C22×C73—42no—
160C22×C73—42no—
161C22×C73—42no—
162C22×C73—42no—
163C22×C73—42no—
164C22×C73—88no—
165C22×C73#44212yes—
166C22×C73C2×C4×D411yescharacteristic
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—42no—
171C4×C73—42no—
172C4×C73#44212yes—
173C4×C73C22×M1611yescharacteristic
174C4×C73#56811yescharacteristic
175C4×C73C2×AGL(1,ℤ/8)12yes—
176#18044—731nomaximal, Sylow
177C2×C73—22no—
178C2×C73—44no—
179C2×C73—82no—
180C2×C73D4×M1611yescharacteristic
181C2×C73C2×#44211yescharacteristic
182C2×C73#295711yescharacteristic
183C2×C4×D8D7311yescharacteristic
184D4×M16—732no—
185#1594—732no—
186#1586—734no—
187#1580—732no—
188#1566—731no—
189#1486—732no—
190#1451—731no—
191C73#1804411yescharacteristic, complemented, Sylow
192C22×D8Dic7311yescharacteristic
193C22×M16—1462no—
194C2×C4×D4C2×D7312yes—
195C4×D8Dic7312yes—
196C4×D8Dic7311yescharacteristic
197C4×D8Dic7311yescharacteristic
198C4×D8C2×D7311yescharacteristic
199C4×D8C2×D7312yes—
200C4×D8C2×D7311yescharacteristic
201#436—734no—
202#435—1464no—
203C8×D4—734no—
204#431—731no—
205C2×#107Dic7311yescharacteristic
206#423—732no—
207C2×#102Dic7312yes—
208#407—1462no—
209C4×M16—731no—
210C2×C4×C8C2×D7311yescharacteristic
211#334—732no—
212#334—732no—
213D4⋊C8—734no—
214D4⋊C8—734no—
215#322—732no—
216#322—732no—
217C22×D4C2×Dic7312yes—
218C23×C4—22no—
219C2×D8—24no—
220C2×D8C2×Dic7312yes—
221C2×D8C2×Dic7311yescharacteristic
222C2×D8C2×Dic7311yescharacteristic
223C2×M16—2928no—
224C2×M16—732no—
225C2×M16—1464no—
226C2×M16—1462no—
227C22×C8—1464no—
228C22×C8C2×Dic7311yescharacteristic
229C22×C8C2×Dic7311yescharacteristic
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×Dic7312yes—
233C4×D4C2×Dic7312yes—
234C4×D4C22×D7312yes—
235C4×D4C22×D7312yes—
236C2×C4⋊C4C2×Dic7312yes—
237C2×C22⋊C4—22no—
238C2×C42C22×D7311yescharacteristic
239#107C2×Dic7312yes—
240#107C2×Dic7311yescharacteristic
241#107C2×Dic7311yescharacteristic
242C4⋊C8—734no—
243C4⋊C8—1462no—
244C4⋊C8—1461no—
245#102C2×Dic7312yes—
246#102C2×Dic7312yes—
247#102C2×Dic7312yes—
248#102C2×Dic7312yes—
249C22⋊C8—1464no—
250C22⋊C8—1464no—
251C22⋊C8—1461no—
252#97—732no—
253C4×C8—732no—
254C4×C8—732no—
255C4×C8C2×Dic7311yescharacteristic
256C4×C8C2×Dic7311yescharacteristic
257C4×C8C22×D7311yescharacteristic
258C4×C8C22×D7311yescharacteristic
259C24—22no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4C22×Dic7312yes—
264C2×D4C22×Dic7312yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4C22×Dic7311yescharacteristic
273C22×C4D4×D7311yescharacteristic
274C22×C4#1192402711yescharacteristic
275D8—24no—
276D8—22no—
277D8#1192401214yescomplemented
278M16—2924no—
279M16—2928no—
280M16—1464no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—2928no—
284C2×C8—734no—
285C2×C8—1464no—
286C2×C8—1462no—
287C2×C8C22×Dic7311yescharacteristic
288C2×C8C22×Dic7311yescharacteristic
289C2×C8C22×Dic7311yescharacteristic
290C2×C8C22×Dic7311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4C22×Dic7312yes—
293C4⋊C4C22×Dic7312yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C42C22×Dic7311yescharacteristic
299C42D4×D7311yescharacteristic
300C42#1192402711yescharacteristic
301C42C23×D7311yescharacteristic
302C23—22no—
303C23—22no—
304C23—44no—
305C23—44no—
306C23—82no—
307C23D4×Dic7311yescharacteristic
308D4—22no—
309D4—22no—
310D4—22no—
311D4—44no—
312D4C2×#1192401214yescomplemented
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4D4×Dic7311yescharacteristic
326C2×C4D4×Dic7311yescharacteristic
327C2×C4D4×Dic7311yescharacteristic
328C2×C4#1455661514yes—
329C2×C4C23×Dic7311yescharacteristic
330C2×C4C2×D4×D7311yescharacteristic
331C2×C4C2×#1192402711yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—2922no—
336C8—2924no—
337C8—1464no—
338C8C2×#1192401212yescomplemented
339C22—21no—
340C22—21no—
341C22—24no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—88no—
348C22#1798869912yes—
349C22C2×D4×Dic7311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4#1798869912yes—
356C4C22×#1192401211yescharacteristic
357C4#1798935411yescharacteristic
358C4C2×#1455661512yes—
359C2—22no—
360C2—44no—
361C2—82no—
362C2D4×#1192401211yescharacteristic
363C2C2×#1798869911yescharacteristic
364C2#2261884811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC2×C4×D8×C73C2
socleC22×C73C2×C4×D4
prime radicalC22×D8×C73C4
cosocleC42×C73C24
hypercenterC2×C4×D8D73
hypocenterC73#18044
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. D4×#11924012C2semidirect
2. #22608845C2semidirect
3. #22608796C2semidirect
4. #22608755C2semidirect
5. #22608687C2semidirect
6. #22608356C2semidirect
7. #22608220C2semidirect
10. #17988665C22semidirect
12. D4×#10596831C22semidirect
17. #17988338C22semidirect
18. #17988338C22semidirect
19. D4⋊#10596831C22semidirect
20. D4⋊#10596831C22semidirect
21. #17988309C22semidirect
22. #17988308C22semidirect
101. D8×C73M16semidirect
129. D4×C73C2×M16semidirect
152. C8×C73C2×M16semidirect
191. C73#18044semidirect
277. D8#11924012semidirect
312. D4C2×#11924012semidirect
338. C8C2×#11924012semidirect

Maximal subgroups

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

subgroupindex
D4×#119240122
#226088452
#226087962
#226087552
#226086872
#226083562
#226082202
C2×C4×D8×C732
#1804473