Finite Group Databasebeta

← #33069280

Subgroups — #33069280

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620869C211yesmaximal, characteristic
2C2×#17988935C211yesmaximal, characteristic
3#22618866C211yesmaximal, characteristic, complemented
4#22609501C214yesmaximal, complemented
5#22609377C214yesmaximal, complemented
6#22608677C211yesmaximal, characteristic, complemented
7#22608616C211yesmaximal, characteristic, complemented
8#22608598C211yesmaximal, characteristic
9C73×#3209C211yesmaximal, characteristic, complemented
10C2×#14556588C2211yescharacteristic
11#17988961—21no—
12#17988935—21no—
13#17988935—21no—
14#17988935—21no—
15#17988935C2212yes—
16#17988935C2212yes—
17#17988787C2214yes—
18#17988737—22no—
19#17988718—21no—
20#17988718C2212yescomplemented
21Q8×#10596831—21no—
22Q8×#10596831C2212yescomplemented
23#17988640C2211yescharacteristic
24#17988614—21no—
25#17988611C2211yescharacteristic
26C2×#14556511—21no—
27C2×#14556511—21no—
28#17988543—21no—
29C2×C4×#10596831C2211yescharacteristic
30#17988339C2212yescomplemented
31#17988337C2212yescomplemented
32#17988333C2214yescomplemented
33#17988330C2214yes—
34#533×C73C2211yescharacteristic
35C2×C4⋊Q8×C73C2211yescharacteristic
36#494×C73—21no—
37#487×C73—21no—
38#462×C73C2214yescomplemented
39#451×C73—22no—
40#431×C73C2211yescharacteristic
41C22×D292—21no—
42C22×D292—21no—
43C22×D292—21no—
44C2×#11924015—24no—
45C2×#11924012—21no—
46C22×#10596831D411yescharacteristic
47C22×#10596831D411yescharacteristic
48#14556588—21no—
49#14556588—21no—
50#14556588—21no—
51#14556588C2312yes—
52C4⋊#10596831—22no—
53C4⋊#10596831—22no—
54C4⋊#10596831—21no—
55C4⋊#10596831C2312yes—
56C4⋊#10596831C2312yes—
57#14556511—22no—
58#14556511—22no—
59#14556511—22no—
60#14556511—22no—
61#14556511—24no—
62C22⋊#10596831—21no—
63#14556502—22no—
64C4×#10596831—22no—
65C4×#10596831—21no—
66C4×#10596831D411yescharacteristic
67C4×#10596831D411yescharacteristic
68C4×#10596831C2311yescharacteristic
69C4×#10596831C2311yescharacteristic
70C22×Q8×C73—21no—
71C2×Q16×C73—24no—
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74C4⋊Q8×C73—21no—
75C4⋊Q8×C73D411yescharacteristic
76C4⋊Q8×C73D411yescharacteristic
77C4⋊Q8×C73C2312yes—
78#125×C73—21no—
79C22⋊Q8×C73—21no—
80C22⋊Q8×C73D412yes—
81C4×Q8×C73—21no—
82C4×Q8×C73C2312yes—
83#117×C73—21no—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73—21no—
86C2×C42×C73C2311yescharacteristic
87C4⋊C8×C73C2312yes—
88#103×C73—24no—
89C22⋊C8×C73—21no—
90C4×C8×C73—21no—
91C23×D73—41no—
92C2×D292—22no—
93C2×D292—22no—
94C2×D292—22no—
95C2×D292—42no—
96C2×D292—42no—
97C2×D292—41no—
98C2×D292—41no—
99C2×D292—41no—
100C2×D292—82no—
101C2×D292—82no—
102#11924015—84no—
103#11924012—42no—
104C2×#10596831—22no—
105C2×#10596831—22no—
106C2×#10596831—22no—
107C2×#10596831—22no—
108C2×#10596831C2×D411yescharacteristic
109C2×#10596831C2×D411yescharacteristic
110C2×#10596831C2×D411yescharacteristic
111C2×#10596831C2×D411yescharacteristic
112C2×Q8×C73—21no—
113C2×Q8×C73—22no—
114C2×Q8×C73—41no—
115C2×Q8×C73—82no—
116C2×Q8×C73C2×D412yes—
117C22×C4×C73—41no—
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73#4111yescharacteristic
121Q16×C73—84no—
122C2×C8×C73—22no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73—21no—
126C4⋊C4×C73—22no—
127C4⋊C4×C73—22no—
128C4⋊C4×C73—22no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C4⋊C4×C73C2×D412yes—
133C22⋊C4×C73—22no—
134C42×C73—22no—
135C42×C73C2×D411yescharacteristic
136C42×C73C2×D411yescharacteristic
137C42×C73#4111yescharacteristic
138C42×C73C2411yescharacteristic
139C23×C73#12111yescharacteristic
140C22×D73—42no—
141C22×D73—82no—
142C22×D73—82no—
143C22×D73—82no—
144Q8×C73—24no—
145Q8×C73—42no—
146Q8×C73—44no—
147Q8×C73—82no—
148C2×C4×C73—22no—
149C2×C4×C73—22no—
150C2×C4×C73—22no—
151C2×C4×C73—22no—
152C2×C4×C73—42no—
153C2×C4×C73—82no—
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#13512yes—
158C2×C4×C73#13712yes—
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C22×D411yescharacteristic
161C2×C4×C73C2×#4111yescharacteristic
162D292—42no—
163D292—42no—
164D292—44no—
165D292—82no—
166D292—82no—
167D292—82no—
168D292—82no—
169C8×C73—42no—
170#10596831—21no—
171#10596831—21no—
172#10596831—21no—
173#10596831—21no—
174#10596831—42no—
175#10596831—42no—
176C22×C73—21no—
177C22×C73—21no—
178C22×C73#46412yes—
179C22×C73C2×#12111yescharacteristic
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—82no—
183C2×D73—164no—
184C4×C73—21no—
185C4×C73—21no—
186C4×C73—42no—
187C4×C73—42no—
188C4×C73—82no—
189C4×C73#46412yes—
190C4×C73#47112yes—
191C4×C73C2×#13511yescharacteristic
192C4×C73C2×#13711yescharacteristic
193C4×C73#57612yes—
194#19214—731nomaximal, Sylow
195C2×C73—22no—
196C2×C73C2×#46411yescharacteristic
197C2×C73#305811yescharacteristic
198C2×C73C2×#47111yescharacteristic
199D73—162no—
200#3209D7311yescharacteristic, complemented
201#3208—731no—
202C2×#486—731no—
203#2960—731no—
204#1702—734no—
205#1682—734no—
206#1564—731no—
207#1549—731no—
208#1545—731no—
209C73#1921411yescharacteristic, complemented, Sylow
210#533C2×D7311yescharacteristic
211C2×C4⋊Q8C2×D7311yescharacteristic
212C2×#127—731no—
213#494—21no—
214#492—1461no—
215#487—21no—
216#486—732no—
217#486—732no—
218#486—1461no—
219#486—1461no—
220#486—1461no—
221#462C2×D7314yescomplemented
222#460—734no—
223#451—22no—
224#450—1462no—
225#446—732no—
226#446—1461no—
227C8×Q8—732no—
228C8×Q8—1461no—
229#431—731no—
230#431C2×D7311yescharacteristic
231#424—1461no—
232#423—731no—
233C2×#102—1461no—
234C2×#102—1461no—
235#405—1461no—
236C2×C4×C8—731no—
237#335—732no—
238#334—732no—
239#332—734no—
240#331—734no—
241C22×Q8—21no—
242C22×D4—1461no—
243C22×D4—1461no—
244C22×D4—1461no—
245C2×Q16—24no—
246C2×SD16—1464no—
247C2×M16—1461no—
248C22×C8—731no—
249C22×C8—731no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22⋊D7311yescharacteristic
254C4⋊Q8C22⋊D7311yescharacteristic
255C4⋊Q8C22×D7312yes—
256#127—732no—
257#127—1461no—
258#127—1461no—
259#127—1461no—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8C22⋊D7312yes—
263C4×Q8—21no—
264C4×Q8C22×D7312yes—
265#117—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D7311yescharacteristic
269C4⋊C8—732no—
270C4⋊C8—732no—
271C4⋊C8—1462no—
272C4⋊C8—1462no—
273C4⋊C8—1461no—
274C4⋊C8C22×D7312yes—
275#103—24no—
276#102—1462no—
277#102—1462no—
278#102—1462no—
279#102—1462no—
280#102—1464no—
281C22⋊C8—21no—
282C22⋊C8—1461no—
283#97—1462no—
284C4×C8—21no—
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—731no—
289C4×C8—1462no—
290C4×C8—1461no—
291C24—2921no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8C2×C22⋊D7312yes—
297C2×D4—2921no—
298C2×D4—2921no—
299C2×D4—2921no—
300C2×D4—2922no—
301C2×D4—2922no—
302C2×D4—5842no—
303C2×D4—5842no—
304C2×D4—1462no—
305C2×D4—1462no—
306C2×D4—1462no—
307C22×C4—41no—
308C22×C4D4×D7311yescharacteristic
309C22×C4C2×C22⋊D7311yescharacteristic
310C22×C4#1192402711yescharacteristic
311Q16—84no—
312SD16—5844no—
313M16—2922no—
314C2×C8—22no—
315C2×C8—731no—
316C2×C8—731no—
317C2×C8—731no—
318C2×C8—731no—
319C2×C8—1462no—
320C2×C8—1462no—
321C2×C8—1462no—
322C2×C8—1462no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4C2×C22⋊D7312yes—
333C22⋊C4—22no—
334C42—22no—
335C42D4×D7311yescharacteristic
336C42C2×C22⋊D7311yescharacteristic
337C42#1192402711yescharacteristic
338C42C23×D7311yescharacteristic
339C23—2922no—
340C23—5842no—
341C23—5842no—
342C23—5842no—
343C23#1455656311yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—44no—
347Q8—82no—
348D4—2922no—
349D4—2924no—
350D4—2922no—
351D4—5842no—
352D4—5842no—
353D4—5842no—
354D4—5842no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4#1455656311yescharacteristic
362C2×C4#1455656311yescharacteristic
363C2×C4#1455656311yescharacteristic
364C2×C4#1455661412yes—
365C2×C4#1455662312yes—
366C2×C4C2×D4×D7311yescharacteristic
367C2×C4C22×C22⋊D7311yescharacteristic
368C2×C4C2×#1192402711yescharacteristic
369C8—42no—
370C8—2922no—
371C8—2922no—
372C8—1461no—
373C8—1461no—
374C8—1461no—
375C8—1461no—
376C22—21no—
377C22—21no—
378C22—5842no—
379C22—5842no—
380C22—5842no—
381C22—11684no—
382C22#1798881112yes—
383C22C2×#1455656311yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—82no—
389C4#1798881112yes—
390C4#1798885212yes—
391C4C2×#1455661411yescharacteristic
392C4C2×#1455662311yescharacteristic
393C4#1798939012yes—
394C2—22no—
395C2—11682no—
396C2C2×#1798881111yescharacteristic
397C2#2261957611yescharacteristic
398C2C2×#1798885211yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3209C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#3209D73
hypocenterC73#19214
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #22618866C2semidirect
4. #22609501C2semidirect
5. #22609377C2semidirect
6. #22608677C2semidirect
7. #22608616C2semidirect
9. C73×#3209C2semidirect
20. #17988718C22semidirect
22. Q8×#10596831C22semidirect
30. #17988339C22semidirect
31. #17988337C22semidirect
32. #17988333C22semidirect
38. #462×C73C22semidirect
200. #3209D73semidirect
209. C73#19214semidirect
221. #462C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226208692
C2×#179889352
#226188662
#226095012
#226093772
#226086772
#226086162
#226085982
C73×#32092
#1921473