Finite Group Databasebeta

← #33068226

Subgroups — #33068226

Order 18688.

802 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620359C211yesmaximal, characteristic, complemented
2#22619931C212yesmaximal, complemented
3#22609296C212yesmaximal, complemented
4#22609286C212yesmaximal, complemented
5#22609272C214yesmaximal, complemented
6#22608679C211yesmaximal, characteristic, complemented
7#22608581C212yesmaximal, complemented
8C2×C8×D4×C73C211yesmaximal, characteristic, complemented
9C2×#14556613—21no—
10C2×#14556613—21no—
11#17989137C2212yes—
12#17988971—21no—
13#17988971—21no—
14#17988967C2211yescharacteristic, complemented
15#17988964C2211yescharacteristic, complemented
16#17988960C2212yescomplemented
17#17988900—22no—
18#17988867C2214yes—
19#17988782C2214yes—
20#17988721—22no—
21C4×#11924013—24no—
22#17988640C2212yes—
23#17988340C2212yescomplemented
24#17988338C2212yescomplemented
25#17988334C2214yescomplemented
26#17988330C2214yescomplemented
27C23×C8×C73—21no—
28C2×C4×D4×C73C2211yescharacteristic
29C8×D4×C73—21no—
30C8×D4×C73—21no—
31C8×D4×C73C2212yes—
32C8×D4×C73C2212yes—
33C2×C4⋊C8×C73C2211yescharacteristic
34C2×C22⋊C8×C73—21no—
35C2×C4×C8×C73C2211yescharacteristic
36C2×#11924028—22no—
37#14556613—44no—
38#14556613—44no—
39C2×Dic292—21no—
40C2×Dic292—21no—
41C2×#11924017—22no—
42C2×#11924017—22no—
43C2×D584—21no—
44C2×D584—21no—
45C2×#11924013—24no—
46#14556591C2312yes—
47#14556588C2312yes—
48#14556579—22no—
49#14556565—22no—
50#14556565—22no—
51C4×D292—22no—
52C4×D292—22no—
53#14556524—24no—
54C4⋊#10596831C2314yes—
55#14556512—24no—
56#14556511—24no—
57C22⋊#10596831—22no—
58C4×#10596831—22no—
59C22×D4×C73D411yescharacteristic
60C23×C4×C73—21no—
61C22×C8×C73—21no—
62C22×C8×C73—22no—
63C22×C8×C73—22no—
64C22×C8×C73—44no—
65C22×C8×C73D411yescharacteristic
66C22×C8×C73D411yescharacteristic
67C4×D4×C73—21no—
68C4×D4×C73—21no—
69C4×D4×C73D412yes—
70C4×D4×C73C2312yes—
71C2×C4⋊C4×C73D411yescharacteristic
72C2×C22⋊C4×C73—21no—
73C2×C42×C73C2311yescharacteristic
74C4⋊C8×C73—21no—
75C4⋊C8×C73C2312yes—
76C22⋊C8×C73—22no—
77C22⋊C8×C73—22no—
78C4×C8×C73D411yescharacteristic
79C4×C8×C73D411yescharacteristic
80C4×C8×C73C2311yescharacteristic
81C4×C8×C73C2311yescharacteristic
82#11924028—44no—
83#11924028—44no—
84C2×Dic146—22no—
85C2×D292—22no—
86C2×D292—22no—
87C2×D292—22no—
88C2×C22⋊D73—42no—
89C2×C4×D73—42no—
90Dic292—42no—
91Dic292—42no—
92#11924017—44no—
93#11924017—44no—
94D584—42no—
95D584—42no—
96#11924013—48no—
97C2×#10596831—24no—
98C4⋊Dic73—22no—
99C4⋊Dic73—22no—
100#11924005—42no—
101#11924005—42no—
102C24×C73—21no—
103C2×D4×C73—24no—
104C2×D4×C73—21no—
105C2×D4×C73C2×D412yes—
106C22×C4×C73—21no—
107C22×C4×C73—22no—
108C22×C4×C73—22no—
109C22×C4×C73—42no—
110C22×C4×C73—42no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73—42no—
117C2×C8×C73—44no—
118C2×C8×C73—44no—
119C2×C8×C73D814yescomplemented
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C4⋊C4×C73—21no—
125C4⋊C4×C73C2×D412yes—
126C22⋊C4×C73—22no—
127C22⋊C4×C73—22no—
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73C2411yescharacteristic
132C23×C73—24no—
133C23×C73—22no—
134C23×C73—42no—
135C23×C73—42no—
136C23×C73#12111yescharacteristic
137C22×D73—42no—
138D4×C73—24no—
139D4×C73—22no—
140D4×C73#13514yescomplemented
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—44no—
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73C2×D812yes—
152C2×C4×C73AGL(1,ℤ/8)12yes—
153C2×C4×C73C22×D411yescharacteristic
154C2×C4×C73C22×D411yescharacteristic
155C2×C4×C73C2×#4111yescharacteristic
156C22⋊D73—84no—
157C2×Dic73—42no—
158D292—42no—
159D292—44no—
160D292—42no—
161D292—82no—
162C4×D73—84no—
163Dic146—42no—
164Dic146—42no—
165C8×C73—21no—
166C8×C73—21no—
167C8×C73—42no—
168C8×C73C2×D812yes—
169C8×C73C2×D812yes—
170#10596831—44no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—24no—
174C22×C73—48no—
175C22×C73—42no—
176C22×C73—42no—
177C22×C73#45912yes—
178C22×C73C2×#12111yescharacteristic
179C2×D73—82no—
180C2×D73—82no—
181C2×D73—82no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—42no—
186C4×C73#45912yes—
187C4×C73C22×D811yescharacteristic
188C4×C73C2×#13512yes—
189C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
190Dic73—82no—
191#19101—731nomaximal, Sylow
192C2×C73—22no—
193C2×C73—44no—
194C2×C73C2×#45911yescharacteristic
195C2×C73#305011yescharacteristic
196C2×C73#306711yescharacteristic
197D73—162no—
198#3147—731no—
199#3100—732no—
200C2×C8×D4D7311yescharacteristic, complemented
201#1669—732no—
202#1667—732no—
203#1665—734no—
204#1564—731no—
205#1540—732no—
206C73#1910111yescharacteristic, complemented, Sylow
207C2×#135—1461no—
208C2×#135—1461no—
209C23×C8—21no—
210#532—732no—
211C2×C4×D4C2×D7311yescharacteristic
212#495—1461no—
213#495—1461no—
214#494—731no—
215#493—731no—
216#492—732no—
217#480—1462no—
218#474—734no—
219#459—734no—
220#447—1462no—
221C4×D8—1464no—
222C8×D4—21no—
223C8×D4—21no—
224C8×D4C2×D7312yes—
225C8×D4C2×D7312yes—
226#431—732no—
227C2×C4⋊C8C2×D7311yescharacteristic
228C2×C22⋊C8—21no—
229C2×C4×C8C2×D7311yescharacteristic
230#335—732no—
231#334—732no—
232#332—734no—
233#331—734no—
234C2×#41—1462no—
235C22×D4D29211yescharacteristic
236C23×C4—21no—
237#135—2924no—
238#135—2924no—
239C2×Q16—1461no—
240C2×Q16—1461no—
241C2×SD16—1462no—
242C2×SD16—1462no—
243C2×D8—1464no—
244C2×D8—1461no—
245C2×D8—1461no—
246C22×C8—21no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—44no—
250C22×C8C22⋊D7311yescharacteristic
251C22×C8C22⋊D7311yescharacteristic
252C4⋊Q8—732no—
253#127—732no—
254#124—1462no—
255#121—1462no—
256#121—1462no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1462no—
260C4×D4—1462no—
261C4×D4D29212yes—
262C4×D4C22×D7312yes—
263C2×C4⋊C4D29211yescharacteristic
264C2×C22⋊C4—21no—
265C2×C42C22×D7311yescharacteristic
266#107—1464no—
267C4⋊C8—21no—
268C4⋊C8—734no—
269C4⋊C8C22×D7312yes—
270#102—1464no—
271#102—1464no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—1462no—
275C4×C8—1462no—
276C4×C8C22⋊D7311yescharacteristic
277C4×C8C22⋊D7311yescharacteristic
278C4×C8C22×D7311yescharacteristic
279C4×C8C22×D7311yescharacteristic
280C24—21no—
281#41—2924no—
282#41—2924no—
283C2×Q8—1462no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—2922no—
287C2×D4—1462no—
288C2×D4—1462no—
289C2×D4—1462no—
290C2×D4C2×D29212yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—2922no—
297C22×C4C2×C22⋊D7311yescharacteristic
298C22×C4C2×D29211yescharacteristic
299C22×C4#1192402811yescharacteristic
300Q16—2922no—
301Q16—2922no—
302SD16—2924no—
303SD16—2924no—
304D8—2922no—
305D8—2928no—
306D8—2922no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—1464no—
313C2×C8#1192401314yescomplemented
314C2×C8C2×C22⋊D7311yescharacteristic
315C2×C8C2×C22⋊D7311yescharacteristic
316C2×C8C2×C22⋊D7311yescharacteristic
317C2×C8C2×C22⋊D7311yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—1462no—
320C4⋊C4—1462no—
321C4⋊C4C2×D29212yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—2922no—
325C22⋊C4—2922no—
326C42C2×C22⋊D7311yescharacteristic
327C42C2×D29211yescharacteristic
328C42#1192402811yescharacteristic
329C42C23×D7311yescharacteristic
330C23—24no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—2922no—
335C23#1455656511yescharacteristic
336Q8—2922no—
337Q8—2922no—
338D4—24no—
339D4—22no—
340D4—2922no—
341D4—2924no—
342D4—2922no—
343D4—5842no—
344D4—5844no—
345D4#1455661314yescomplemented
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—2922no—
354C2×C4—5844no—
355C2×C4#1455656511yescharacteristic
356C2×C4#1455656511yescharacteristic
357C2×C4#1455656511yescharacteristic
358C2×C4C2×#1192401312yes—
359C2×C4#1455661612yes—
360C2×C4C22×C22⋊D7311yescharacteristic
361C2×C4C22×D29211yescharacteristic
362C2×C4C2×#1192402811yescharacteristic
363C8—21no—
364C8—21no—
365C8—42no—
366C8—2924no—
367C8C2×#1192401312yes—
368C8C2×#1192401312yes—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—48no—
373C22—42no—
374C22—42no—
375C22—5842no—
376C22—5842no—
377C22—5842no—
378C22#1798878212yes—
379C22C2×#1455656511yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—5842no—
385C4#1798878212yes—
386C4C22×#1192401311yescharacteristic
387C4C2×#1455661312yes—
388C4C2×#1455661611yescharacteristic
389C2—22no—
390C2—44no—
391C2—11682no—
392C2C2×#1798878211yescharacteristic
393C2#2261951211yescharacteristic
394C2#2261965111yescharacteristic
subgroupG/H
centerC2×C4#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×C8×D4×C73C2
socleC22×C73C2×#121
hypercenterC2×C8×D4D73
hypocenterC73#19101
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #22620359C2semidirect
2. #22619931C2semidirect
3. #22609296C2semidirect
4. #22609286C2semidirect
5. #22609272C2semidirect
6. #22608679C2semidirect
7. #22608581C2semidirect
8. C2×C8×D4×C73C2semidirect
14. #17988967C22semidirect
15. #17988964C22semidirect
16. #17988960C22semidirect
23. #17988340C22semidirect
24. #17988338C22semidirect
25. #17988334C22semidirect
26. #17988330C22semidirect
119. C2×C8×C73D8semidirect
140. D4×C73#135semidirect
200. C2×C8×D4D73semidirect
206. C73#19101semidirect
313. C2×C8#11924013semidirect
345. D4#14556613semidirect

Maximal subgroups

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

subgroupindex
#226203592
#226199312
#226092962
#226092862
#226092722
#226086792
#226085812
C2×C8×D4×C732
#1910173