Finite Group Databasebeta

← #10696670

Subgroups — #10696670

Order 768.

514 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 87 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#83216C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#79064C211yesmaximal, characteristic
4C3×#1966C211yesmaximal, characteristic, complemented
5#69859C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#10108—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#101—22no—
14C3×C4×M16C2211yescharacteristic
15#5135—22no—
16#5093—21no—
17#5089—22no—
18#5088—21no—
19#5088—21no—
20#5084—22no—
21#5080—21no—
22#4968—22no—
23#4966—22no—
24#4925C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4843—21no—
27#4843—21no—
28#4842—21no—
29#4842—21no—
30#4464—22no—
31#4459—22no—
32#3253—31no—
33#3227—31no—
34#3141—31no—
35#1966D311yescharacteristic, complemented
36#1525—31no—
37#1484—31no—
38#1299—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×C3×M16—21no—
43C2×C3×M16D412yes—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55C2×#208—21no—
56C2×D24—22no—
57#875—21no—
58#875—21no—
59#874D411yescharacteristic
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×#101—44no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#830—21no—
73#821—22no—
74#821—22no—
75#817—22no—
76#810—24no—
77#809—24no—
78#809—24no—
79#808—24no—
80C4⋊(C3⋊C8)D412yes—
81#803D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#532—31no—
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87#498—61no—
88#489—62no—
89#487—61no—
90#486—61no—
91#486—61no—
92#481—62no—
93#476—61no—
94#476—61no—
95#476—62no—
96#469—62no—
97#459—61no—
98#459—61no—
99#431—31no—
100#423C2×D311yescharacteristic
101#417—62no—
102C2×#101—22no—
103C4×M16—31no—
104C4×M16C2×D311yescharacteristic
105#329—62no—
106D4⋊C8—62no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—41no—
116C2×#67—41no—
117#235—44no—
118#235—82no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×M16—22no—
130C3×M16—41no—
131C3×M16—42no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150#208—42no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153D24—82no—
154C2×#41—61no—
155C2×#41—61no—
156C22×Q8—21no—
157C2×D8—62no—
158C2×M16—21no—
159C2×M16—61no—
160C2×M16#6712yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167#127—31no—
168#127—31no—
169#124—61no—
170#124—61no—
171#121—61no—
172#121—61no—
173#121—61no—
174#121—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4D1212yes—
181C2×C42C22×D311yescharacteristic
182#106—64no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187#103—64no—
188#102—64no—
189#102—64no—
190#102—62no—
191#102—62no—
192#101—44no—
193C22⋊C8—61no—
194#97—21no—
195#97—32no—
196C4×C8—21no—
197C4×C8—61no—
198C4×C8—61no—
199C23×C3#12111yescharacteristic
200C22×D3—41no—
201C22×D3—41no—
202C3×Q8—82no—
203C3×Q8—81no—
204C3×Q8—81no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—44no—
210C2×C3×C4—42no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217#67—82no—
218#67—82no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221D12—42no—
222D12—41no—
223D12—41no—
224D12—42no—
225D12—41no—
226D12—41no—
227D12—42no—
228D12—81no—
229C4×D3—82no—
230C4×D3—82no—
231Dic6—42no—
232Dic6—81no—
233C3×C8—42no—
234C3×C8—42no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237#41—124no—
238#41—242no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4—121no—
253C22×C4—22no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4C2×D1211yescharacteristic
257C22×C4#23511yescharacteristic
258C22×C4C2×#6711yescharacteristic
259D8—242no—
260M16—22no—
261M16—41no—
262M16—42no—
263M16—122no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—62no—
267C2×C8—62no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42D3×D411yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—82no—
298C3×C4#45311yescharacteristic
299C3×C4#45412yes—
300C3×C4#45411yescharacteristic
301C3×C4#47112yes—
302Dic3—81no—
303Dic3—81no—
304C23—121no—
305C23—121no—
306C23#93011yescharacteristic
307Q8—82no—
308Q8—81no—
309Q8—81no—
310Q8—122no—
311Q8—241no—
312D4—122no—
313D4—122no—
314D4—122no—
315D4—121no—
316D4—121no—
317D4—121no—
318D4—121no—
319D4—241no—
320D4—242no—
321D4—242no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—242no—
331C2×C4—242no—
332C2×C4#88211yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#93711yescharacteristic
337C2×C4#94011yescharacteristic
338C8—42no—
339C8—42no—
340C8—122no—
341C8—122no—
342C2×C3—41no—
343C2×C3#200311yescharacteristic
344C2×C3#200611yescharacteristic
345C2×C3#201911yescharacteristic
346D3—161no—
347D3—161no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—241no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22#496011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—241no—
363C4—241no—
364C4#475312yes—
365C4#475412yes—
366C4#506211yescharacteristic
367C4#507711yescharacteristic
368C3#1010811yescharacteristic, complemented, Sylow
369C2—41no—
370C2—481no—
371C2—481no—
372C2#6966411yescharacteristic
373C2#7011611yescharacteristic
374C2#7011711yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1966C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1966D3
hypocenterC3#10108
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
1. #83216C2semidirect
2. #83199C2semidirect
4. C3×#1966C2semidirect
5. #69859C2semidirect
6. #69853C2semidirect
7. #67710C2semidirect
35. #1966D3semidirect
368. C3#10108semidirect

Maximal subgroups

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

subgroupindex
#832162
#831992
#790642
C3×#19662
#698592
#698532
#677102
#101083