Finite Group Databasebeta

← #10849659

Subgroups — #10849659

Order 768.

1072 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 175 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#86190C211yesmaximal, characteristic
2C3×#2641C211yesmaximal, characteristic, complemented
3#83017C211yesmaximal, characteristic, complemented
4#83015C211yesmaximal, characteristic, complemented
5#71766C211yesmaximal, characteristic, complemented
6#69665C214yesmaximal, complemented
7#69642C214yesmaximal, complemented
8#69441C211yesmaximal, characteristic, complemented
9#69430C211yesmaximal, characteristic, complemented
10#25690—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#125C2211yescharacteristic
13#5806—24no—
14#5805C2212yes—
15#5798C2212yes—
16#5719—22no—
17#5718—22no—
18#5716—22no—
19#5713—22no—
20C4×#235C2211yescharacteristic
21#5470C2211yescharacteristic
22C3×#394—24no—
23C3×#390C2214yes—
24C3×#376C2211yescharacteristic
25C2×#833C2214yes—
26#5066—22no—
27#5065—22no—
28#5064—22no—
29#5063—22no—
30#4966C2212yes—
31#4964C2212yes—
32#4964C2214yes—
33C2×#805C2212yes—
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4523C2212yes—
37#4523C2214yes—
38#3508—31no—
39#3166—31no—
40#3080—31no—
41#2641D311yescharacteristic, complemented
42#2037—34no—
43#1995—31no—
44#1965—31no—
45#1856—34no—
46#1759—31no—
47C22×C3×D4D414yes—
48C22×C3×D4C2312yes—
49C2×#237—24no—
50C2×#235C2311yescharacteristic
51C3×#127—22no—
52C3×#127—21no—
53C3×#127—21no—
54C3×#125—22no—
55C3×#125—21no—
56C3×#125—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—28no—
60C2×C3×C42C2312yes—
61C2×C3×C42C2311yescharacteristic
62#953—24no—
63C2×#217—24no—
64#940—22no—
65#939—28no—
66#938—24no—
67#938—24no—
68#936—24no—
69#936—22no—
70#935—28no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73D4×Dic3—24no—
74C4×#67—22no—
75#926C2312yes—
76#926C2311yescharacteristic
77#924—22no—
78C2×#208C2312yes—
79C2×#208C2312yes—
80C4×D12D412yes—
81#872—21no—
82C42×D3—21no—
83#870D412yes—
84#869D412yes—
85C4×Dic6—22no—
86C4×Dic6D412yes—
87C3×#95—22no—
88#833—28no—
89#810—22no—
90#809—22no—
91#808—22no—
92#808—24no—
93#807—22no—
94#807—24no—
95#805—24no—
96C4⋊(C3⋊C8)—22no—
97#803—21no—
98C4×C3⋊C8—21no—
99#559—62no—
100#550—62no—
101#546—64no—
102#540—62no—
103#535—32no—
104#533—31no—
105#532—62no—
106C2×#127C2×D311yescharacteristic
107C2×#125C2×D311yescharacteristic
108#518—32no—
109C4×#41—31no—
110#501—62no—
111#499—62no—
112#479—62no—
113#478—62no—
114#428—32no—
115#428—34no—
116#423—31no—
117C2×#104—32no—
118#417—32no—
119C2×#100—34no—
120C4×M16—31no—
121#394—24no—
122#390C2×D314yes—
123#376C2×D311yescharacteristic
124#339—32no—
125#339—34no—
126C24×C3C2×D414yes—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—84no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2411yescharacteristic
138C2×#67—21no—
139C2×#67—28no—
140C22×Dic3—28no—
141#237—88no—
142#235—22no—
143#235—22no—
144C2×D12C2×D411yescharacteristic
145C2×C4×D3—22no—
146C2×C4×D3—21no—
147C2×Dic6—22no—
148C2×Dic6C2×D411yescharacteristic
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—42no—
153C3×C22⋊C4—88no—
154C3×C42—22no—
155C3×C42—22no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D412yes—
159#217—28no—
160#217—28no—
161#217—22no—
162#217—21no—
163(C2×D3)⋊C4—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3C2×D412yes—
166C4⋊Dic3C2×D414yes—
167Dic3⋊C4—24no—
168Dic3⋊C4—22no—
169C4×Dic3—22no—
170C4×Dic3—22no—
171C4×Dic3—21no—
172#208—22no—
173#208—24no—
174#208#4114yescomplemented
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×#41—31no—
178C2×#41—64no—
179C22×D4#6714yes—
180C22×D4C22×D312yes—
181C2×M16—32no—
182C2×M16—32no—
183C4⋊Q8—32no—
184#127—21no—
185#127—21no—
186#127—22no—
187#127—62no—
188#125—22no—
189#125—21no—
190#125—21no—
191#125—32no—
192#124—62no—
193#124—64no—
194#123—68no—
195C22⋊Q8—62no—
196#121—64no—
197#121—64no—
198#121—68no—
199#120—64no—
200C4×Q8—32no—
201C4×Q8—62no—
202C4×D4—32no—
203C4×D4—64no—
204C4×D4—64no—
205C4×D4—64no—
206C4×D4—62no—
207#117—31no—
208#117—32no—
209#117—61no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22×D312yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—64no—
214C2×C42—61no—
215C2×C42C22×D312yes—
216C2×C42C22×D311yescharacteristic
217#107—62no—
218#107—64no—
219#106—62no—
220#106—64no—
221C4⋊C8—62no—
222#104—64no—
223#103—62no—
224#102—62no—
225#100—68no—
226#97—61no—
227C4×C8—61no—
228#95—22no—
229C23×C3—24no—
230C23×C3—28no—
231C23×C3—48no—
232C23×C3C22×D411yescharacteristic
233C22×D3—21no—
234C3×D4—48no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—44no—
238C3×D4—84no—
239C2×C3×C4—42no—
240C2×C3×C4—44no—
241C2×C3×C4—44no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4#12012yes—
245C2×C3×C4#12012yes—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C42−1+412yes—
250#67—42no—
251#67—88no—
252C2×Dic3—21no—
253C2×Dic3—22no—
254C2×Dic3—24no—
255C2×Dic3—88no—
256D12—21no—
257D12—21no—
258C4×D3—44no—
259C4×D3—42no—
260Dic6—21no—
261Dic6—21no—
262Dic6—44no—
263C3⋊C8—24no—
264C3⋊C8—42no—
265C24C2×#6714yes—
266#41—62no—
267#41—62no—
268#41—248no—
269C2×Q8—31no—
270C2×Q8—62no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—31no—
278C2×D4—61no—
279C2×D4—68no—
280C2×D4—84no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—68no—
284C22×C4—62no—
285C22×C4—61no—
286C22×C4C2×#6712yes—
287C22×C4C23×D311yescharacteristic
288M16—34no—
289M16—62no—
290M16—64no—
291C2×C8—62no—
292C2×C8—62no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—64no—
302C22⋊C4—68no—
303C22⋊C4—68no—
304C22⋊C4—61no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—88no—
308C42—22no—
309C42—22no—
310C42—62no—
311C42—61no—
312C42—62no—
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C42D3×D412yes—
316C22×C3—22no—
317C22×C3—44no—
318C22×C3—48no—
319C22×C3—88no—
320C22×C3C2×#12011yescharacteristic
321C22×C3#54712yes—
322C2×D3—41no—
323C2×D3—42no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—82no—
327C3×C4—82no—
328C3×C4#53211yescharacteristic
329C3×C4#53611yescharacteristic
330C3×C4#54812yes—
331Dic3—41no—
332Dic3—44no—
333Dic3—42no—
334C23—24no—
335C23—28no—
336C23—48no—
337C23—61no—
338C23C22×#6711yescharacteristic
339Q8—61no—
340Q8—61no—
341Q8—124no—
342D4—48no—
343D4—42no—
344D4—42no—
345D4—44no—
346D4—61no—
347D4—61no—
348D4—84no—
349D4—122no—
350D4—248no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—61no—
357C2×C4—62no—
358C2×C4—64no—
359C2×C4—124no—
360C2×C4—122no—
361C2×C4—248no—
362C2×C4#93712yes—
363C2×C4#93712yes—
364C2×C4C2×D3×D411yescharacteristic
365C2×C4C2×D3×D411yescharacteristic
366C2×C4D3×#4112yes—
367C2×C4#101012yes—
368C8—64no—
369C8—122no—
370C2×C3—22no—
371C2×C3—84no—
372C2×C3#243211yescharacteristic
373C2×C3#301912yes—
374D3—81no—
375C22—22no—
376C22—44no—
377C22—48no—
378C22—88no—
379C22—121no—
380C22—122no—
381C22C2×#93711yescharacteristic
382C22#582412yes—
383C4—22no—
384C4—22no—
385C4—82no—
386C4—82no—
387C4—121no—
388C4—122no—
389C4—124no—
390C4#566811yescharacteristic
391C4#567511yescharacteristic
392C4#567712yes—
393C3#2569011yescharacteristic, complemented, Sylow
394C2—22no—
395C2—84no—
396C2—241no—
397C2#8295911yescharacteristic
398C2#8339412yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2641C2
socleC22×C3C2×#120
prime radical#86190C2
hypercenter#2641D3
hypocenterC3#25690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2641

Complements

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

NH ≅ G/Ntype
2. C3×#2641C2semidirect
3. #83017C2semidirect
4. #83015C2semidirect
5. #71766C2semidirect
6. #69665C2semidirect
7. #69642C2semidirect
8. #69441C2semidirect
9. #69430C2semidirect
41. #2641D3semidirect
174. #208#41semidirect
393. C3#25690semidirect

Maximal subgroups

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

subgroupindex
#861902
C3×#26412
#830172
#830152
#717662
#696652
#696422
#694412
#694302
#256903