Finite Group Databasebeta

← #33069262

Subgroups — #33069262

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 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#22609473C214yesmaximal, complemented
5#22609349C214yesmaximal, complemented
6#22608677C211yesmaximal, characteristic, complemented
7#22608612C211yesmaximal, characteristic, complemented
8#22608598C211yesmaximal, characteristic
9C73×#3230C211yesmaximal, 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#17988336C2214yescomplemented
33#17988330C2214yes—
34#533×C73C2211yescharacteristic
35C2×C4⋊Q8×C73C2211yescharacteristic
36#500×C73—21no—
37#487×C73—21no—
38#484×C73—22no—
39#477×C73C2214yescomplemented
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—
71C4⋊Q8×C73—21no—
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74C4⋊Q8×C73D411yescharacteristic
75C4⋊Q8×C73D411yescharacteristic
76C4⋊Q8×C73C2312yes—
77#125×C73—21no—
78C22⋊Q8×C73—21no—
79C22⋊Q8×C73D412yes—
80C4×Q8×C73—21no—
81C4×Q8×C73C2312yes—
82#117×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C4⋊C4×C73—21no—
85C2×C42×C73C2311yescharacteristic
86#107×C73—24no—
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
121C2×C8×C73—22no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73—22no—
126C4⋊C4×C73—22no—
127C4⋊C4×C73—22no—
128C4⋊C4×C73—22no—
129C4⋊C4×C73—41no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73C2×D412yes—
132C22⋊C4×C73—22no—
133C42×C73—22no—
134C42×C73C2×D411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73#4111yescharacteristic
137C42×C73C2411yescharacteristic
138C23×C73#12111yescharacteristic
139C22×D73—42no—
140C22×D73—82no—
141C22×D73—82no—
142C22×D73—82no—
143Q8×C73—24no—
144Q8×C73—42no—
145Q8×C73—82no—
146Q8×C73—82no—
147C2×C4×C73—22no—
148C2×C4×C73—22no—
149C2×C4×C73—22no—
150C2×C4×C73—22no—
151C2×C4×C73—42no—
152C2×C4×C73—82no—
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#13512yes—
157C2×C4×C73#13712yes—
158C2×C4×C73C22×D411yescharacteristic
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C2×#4111yescharacteristic
161D292—42no—
162D292—42no—
163D292—44no—
164D292—82no—
165D292—82no—
166D292—82no—
167D292—82no—
168C8×C73—42no—
169#10596831—21no—
170#10596831—21no—
171#10596831—21no—
172#10596831—21no—
173#10596831—42no—
174#10596831—42no—
175C22×C73—21no—
176C22×C73—21no—
177C22×C73#46412yes—
178C22×C73C2×#12111yescharacteristic
179C2×D73—82no—
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—164no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—42no—
186C4×C73—42no—
187C4×C73—82no—
188C4×C73#46412yes—
189C4×C73#47212yes—
190C4×C73C2×#13511yescharacteristic
191C4×C73C2×#13711yescharacteristic
192C4×C73#57712yes—
193#19212—731nomaximal, Sylow
194C2×C73—22no—
195C2×C73C2×#46411yescharacteristic
196C2×C73#305711yescharacteristic
197C2×C73C2×#47211yescharacteristic
198D73—162no—
199#3230D7311yescharacteristic, complemented
200#3208—731no—
201C2×#486—731no—
202#2960—731no—
203#1698—734no—
204#1678—734no—
205#1564—731no—
206#1548—731no—
207#1545—731no—
208C73#1921211yescharacteristic, complemented, Sylow
209#533C2×D7311yescharacteristic
210C2×C4⋊Q8C2×D7311yescharacteristic
211C2×#127—731no—
212#500—21no—
213#492—1461no—
214#487—21no—
215#486—732no—
216#486—732no—
217#486—1461no—
218#486—1461no—
219#486—1461no—
220#484—22no—
221#477C2×D7314yescomplemented
222#460—734no—
223#450—1462no—
224#446—732no—
225#446—1461no—
226C8×Q8—732no—
227C8×Q8—1461no—
228#431—731no—
229#431C2×D7311yescharacteristic
230#424—1461no—
231#423—731no—
232C2×#102—1461no—
233C2×#102—1461no—
234#405—1461no—
235C2×C4×C8—731no—
236#335—732no—
237#334—732no—
238#333—734no—
239#331—734no—
240C22×Q8—21no—
241C22×D4—1461no—
242C22×D4—1461no—
243C22×D4—1461no—
244C2×SD16—1464no—
245C2×M16—1461no—
246C22×C8—731no—
247C22×C8—731no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D7311yescharacteristic
252C4⋊Q8C22⋊D7311yescharacteristic
253C4⋊Q8C22×D7312yes—
254#127—732no—
255#127—1461no—
256#127—1461no—
257#127—1461no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8C22⋊D7312yes—
261C4×Q8—21no—
262C4×Q8C22×D7312yes—
263#117—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C22×D7311yescharacteristic
267#107—24no—
268C4⋊C8—732no—
269C4⋊C8—732no—
270C4⋊C8—1462no—
271C4⋊C8—1462no—
272C4⋊C8—1461no—
273C4⋊C8C22×D7312yes—
274#103—24no—
275#102—1462no—
276#102—1462no—
277#102—1462no—
278#102—1462no—
279#102—1464no—
280C22⋊C8—21no—
281C22⋊C8—1461no—
282#97—1462no—
283C4×C8—21no—
284C4×C8—731no—
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—1462no—
289C4×C8—1461no—
290C24—2921no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×C22⋊D7312yes—
296C2×D4—2921no—
297C2×D4—2921no—
298C2×D4—2921no—
299C2×D4—2922no—
300C2×D4—2922no—
301C2×D4—5842no—
302C2×D4—5842no—
303C2×D4—1462no—
304C2×D4—1462no—
305C2×D4—1462no—
306C22×C4—41no—
307C22×C4D4×D7311yescharacteristic
308C22×C4C2×C22⋊D7311yescharacteristic
309C22×C4#1192402711yescharacteristic
310SD16—5844no—
311M16—2922no—
312C2×C8—22no—
313C2×C8—731no—
314C2×C8—731no—
315C2×C8—731no—
316C2×C8—731no—
317C2×C8—1462no—
318C2×C8—1462no—
319C2×C8—1462no—
320C2×C8—1462no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4C2×C22⋊D7312yes—
331C22⋊C4—22no—
332C42—22no—
333C42D4×D7311yescharacteristic
334C42C2×C22⋊D7311yescharacteristic
335C42#1192402711yescharacteristic
336C42C23×D7311yescharacteristic
337C23—2922no—
338C23—5842no—
339C23—5842no—
340C23—5842no—
341C23#1455656311yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—82no—
345Q8—82no—
346D4—2922no—
347D4—2924no—
348D4—2922no—
349D4—5842no—
350D4—5842no—
351D4—5842no—
352D4—5842no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4#1455656311yescharacteristic
360C2×C4#1455656311yescharacteristic
361C2×C4#1455656311yescharacteristic
362C2×C4#1455661412yes—
363C2×C4#1455662312yes—
364C2×C4C2×D4×D7311yescharacteristic
365C2×C4C22×C22⋊D7311yescharacteristic
366C2×C4C2×#1192402711yescharacteristic
367C8—42no—
368C8—2922no—
369C8—2922no—
370C8—1461no—
371C8—1461no—
372C8—1461no—
373C8—1461no—
374C22—21no—
375C22—21no—
376C22—5842no—
377C22—5842no—
378C22—5842no—
379C22—11684no—
380C22#1798881112yes—
381C22C2×#1455656311yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4#1798881112yes—
388C4#1798885512yes—
389C4C2×#1455661411yescharacteristic
390C4C2×#1455662311yescharacteristic
391C4#1798939512yes—
392C2—22no—
393C2—11682no—
394C2C2×#1798881111yescharacteristic
395C2#2261957211yescharacteristic
396C2C2×#1798885511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3230C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#3230D73
hypocenterC73#19212
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #22618866C2semidirect
4. #22609473C2semidirect
5. #22609349C2semidirect
6. #22608677C2semidirect
7. #22608612C2semidirect
9. C73×#3230C2semidirect
20. #17988718C22semidirect
22. Q8×#10596831C22semidirect
30. #17988339C22semidirect
31. #17988337C22semidirect
32. #17988336C22semidirect
39. #477×C73C22semidirect
199. #3230D73semidirect
208. C73#19212semidirect
221. #477C2×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
#226094732
#226093492
#226086772
#226086122
#226085982
C73×#32302
#1921273