Finite Group Databasebeta

← #10897670

Subgroups — #10897670

Order 768.

646 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84000C211yesmaximal, characteristic, complemented
2#79071C211yesmaximal, characteristic, complemented
3C3×#2053C211yesmaximal, characteristic, complemented
4#72121C211yesmaximal, characteristic
5#70539C211yesmaximal, characteristic, complemented
6#70293C212yesmaximal, complemented
7#70288C212yesmaximal, complemented
8#70055C212yesmaximal, complemented
9#70044C212yesmaximal, complemented
10#68715C211yesmaximal, characteristic, complemented
11#68696C211yesmaximal, characteristic, complemented
12#29387—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14#5822C2212yes—
15#5819C2212yes—
16#5522C2211yescharacteristic
17C3×#424C2211yescharacteristic
18C3×#416C2212yes—
19C3×#410C2212yes—
20C2×#837C2212yes—
21#5234C2212yes—
22#5197—22no—
23#5196—22no—
24#5129C2211yescharacteristic
25#5125C2211yescharacteristic
26Dic3×M16C2211yescharacteristic
27#5110C2211yescharacteristic
28C2×#815C2211yescharacteristic
29#5005C2212yes—
30#4999C2211yescharacteristic
31#4999C2212yes—
32#4840—22no—
33#4835—22no—
34C3×H(3,ℤ/4)C2211yescharacteristic
35C3⋊#338C2211yescharacteristic
36#4552C2212yes—
37#4533C2212yes—
38#4533C2212yes—
39#4532C2212yes—
40#2956—31no—
41#2923—31no—
42#2103—31no—
43#2098—31no—
44#2053D311yescharacteristic, complemented
45#2017—31no—
46#1938—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C2×#235C2311yescharacteristic
53C3×#137—22no—
54C3×AGL(1,ℤ/8)—22no—
55C3×#135—22no—
56C2×C3×M16C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C3×#117C2312yes—
59C3×#117C2311yescharacteristic
60#951—22no—
61#950—42no—
62#949—22no—
63#948—24no—
64#948—42no—
65#942—22no—
66#932—22no—
67C2×C4×Dic3C2311yescharacteristic
68C2×#208—42no—
69C2×#208C2312yes—
70C2×#208C2312yes—
71C2×#208C2311yescharacteristic
72C22×C3⋊C8C2311yescharacteristic
73#893—21no—
74#887—21no—
75#67⋊C4—22no—
76#872—21no—
77C42×D3—21no—
78C3×C4⋊C8—22no—
79C3×#103—22no—
80C3×#102—22no—
81C3×#99—24no—
82#837—22no—
83#837—22no—
84#830—42no—
85(C2×D3)⋊C8—22no—
86(C2×D3)⋊C8—22no—
87#819—22no—
88#815—21no—
89#815—21no—
90#815D412yes—
91C8×Dic3D412yes—
92#805—42no—
93C4⋊(C3⋊C8)—22no—
94C4⋊(C3⋊C8)—42no—
95#575—32no—
96#575C2×D311yescharacteristic
97#568—32no—
98C4×#41—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#432—62no—
103#430—31no—
104#424—31no—
105#424—32no—
106#424C2×D311yescharacteristic
107#421—32no—
108C2×#104—32no—
109#416C2×D312yes—
110#410C2×D312yes—
111#408—31no—
112#408—31no—
113#408—32no—
114C4×M16—31no—
115C2×#97—31no—
116#344—32no—
117#344—32no—
118#341—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C2×C3×Q8#4112yes—
127C2×C3×D4—22no—
128C2×C3×D4#4112yes—
129C22×C3×C4—22no—
130C22×C3×C4C2411yescharacteristic
131C2×#67—21no—
132C22×Dic3—21no—
133#235—22no—
134#235—22no—
135C2×D12#4111yescharacteristic
136C2×C4×D3—21no—
137C2×C4×D3—22no—
138C2×Dic6#4111yescharacteristic
139C3×Q16—22no—
140C3×SD16—24no—
141C3×D8—22no—
142C3×M16—22no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8C2×D412yes—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C22⋊C4#4112yes—
150C3×C42—21no—
151C3×C42—22no—
152#216—42no—
153#215—42no—
154#214—42no—
155#213—42no—
156(C2×D3)⋊C4—22no—
157Dic3⋊C4—22no—
158C4×Dic3—22no—
159C4×Dic3—21no—
160C4×Dic3C2×D411yescharacteristic
161C4×Dic3C2×D411yescharacteristic
162#208—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166#208—22no—
167#208—44no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—44no—
173C2×C3⋊C8C2×D412yes—
174C2×#41—31no—
175C2×#41C22×D312yes—
176#137—22no—
177#137—62no—
178#137—62no—
179AGL(1,ℤ/8)—22no—
180AGL(1,ℤ/8)—62no—
181AGL(1,ℤ/8)—62no—
182#135—22no—
183#135—122no—
184#131—64no—
185#131—122no—
186C2×M16—31no—
187C2×M16—32no—
188C2×M16—32no—
189C2×M16—122no—
190C2×M16C22×D311yescharacteristic
191C22×C8—31no—
192C22×C8C22×D311yescharacteristic
193C4×Q8—61no—
194C4×D4—61no—
195C4×D4—62no—
196#117—61no—
197#117C22×D312yes—
198#117C22×D311yescharacteristic
199C2×C42—31no—
200C2×C42—61no—
201#108—62no—
202C4⋊C8—22no—
203C4⋊C8—62no—
204C4⋊C8—122no—
205#104—62no—
206#104—62no—
207#104—122no—
208#103—22no—
209#102—22no—
210#99—24no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—32no—
215#97—61no—
216#97—61no—
217C4×C8—32no—
218C23×C3—22no—
219C23×C3C2×#4111yescharacteristic
220C22×D3—21no—
221C3×Q8—22no—
222C3×Q8—22no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—42no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4C22×D411yescharacteristic
233C2×C3×C4C2×#4112yes—
234C2×C3×C4C2×#4111yescharacteristic
235C2×C3×C4H(5,𝔽2)12yes—
236#67—42no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—81no—
241D12—22no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—82no—
245Dic6—22no—
246C3×C8—22no—
247C3×C8—41no—
248C3⋊C8—22no—
249C3⋊C8—41no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252#41—22no—
253#41—22no—
254#41—22no—
255#41—22no—
256#41—62no—
257#41—62no—
258C2×Q8—31no—
259C2×Q8#23712yes—
260C2×D4—22no—
261C2×D4—31no—
262C2×D4—61no—
263C2×D4#23712yes—
264C22×C4—22no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4C23×D311yescharacteristic
269Q16—22no—
270Q16—122no—
271SD16—24no—
272SD16—122no—
273SD16—122no—
274D8—22no—
275D8—122no—
276M16—22no—
277M16—62no—
278M16—62no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—124no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—32no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—124no—
291C2×C8D3×D412yes—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—62no—
295C22⋊C4—22no—
296C22⋊C4—62no—
297C22⋊C4#23712yes—
298C42—21no—
299C42—22no—
300C42—31no—
301C42—31no—
302C42—62no—
303C42—61no—
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—82no—
307C22×C3#52511yescharacteristic
308C22×C3#55312yes—
309C2×D3—42no—
310C2×D3—41no—
311C3×C4—42no—
312C3×C4—42no—
313C3×C4—82no—
314C3×C4#52511yescharacteristic
315C3×C4#53511yescharacteristic
316C3×C4#55112yes—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—22no—
321C23—61no—
322C23C2×#23711yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—62no—
326D4—22no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—62no—
331D4—122no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—242no—
343C2×C4—241no—
344C2×C4C2×D3×D411yescharacteristic
345C2×C4C2×#23712yes—
346C2×C4D3×#4111yescharacteristic
347C2×C4#100912yes—
348C8—22no—
349C8—41no—
350C8—62no—
351C8—122no—
352C8—121no—
353C8—122no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—82no—
357C2×C3#259811yescharacteristic
358D3—81no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—122no—
363C22—121no—
364C22#560611yescharacteristic
365C22#561812yes—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4#558611yescharacteristic
373C4#565812yes—
374C4#567411yescharacteristic
375C3#2938711yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—82no—
379C2—241no—
380C2#7696211yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2053C2
socleC2×C3#2598
prime radical#72121C2
hypercenter#2053D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2053

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #79071C2semidirect
3. C3×#2053C2semidirect
5. #70539C2semidirect
6. #70293C2semidirect
7. #70288C2semidirect
8. #70055C2semidirect
9. #70044C2semidirect
10. #68715C2semidirect
11. #68696C2semidirect
44. #2053D3semidirect
375. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#840002
#790712
C3×#20532
#721212
#705392
#702932
#702882
#700552
#700442
#687152
#686962
#293873