Finite Group Databasebeta

← #10696686

Subgroups — #10696686

Order 768.

498 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 89 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#79268C211yesmaximal, characteristic, complemented
2#79064C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic, complemented
4#70761C211yesmaximal, characteristic, complemented
5#70524C211yesmaximal, characteristic, complemented
6#69859C211yesmaximal, characteristic, complemented
7C3×#1299C211yesmaximal, characteristic, complemented
8#10110—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5135—22no—
16#5133—22no—
17C4×#208C2211yescharacteristic
18#4846—22no—
19#4843—21no—
20#4843—21no—
21#4842—21no—
22#4842—21no—
23#4736—22no—
24#4714—22no—
25#4694—21no—
26#4694—21no—
27#4682—21no—
28#4681—21no—
29#4554—22no—
30#4483—22no—
31#4464—22no—
32#3253—31no—
33#3226—31no—
34#3141—31no—
35#1997—31no—
36#1525—31no—
37#1484—31no—
38#1299D311yescharacteristic, complemented
39C2×#235—21no—
40C2×#235—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C2×C3×M16D412yes—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D412yes—
51C2×#208—21no—
52C2×D24—22no—
53#875—21no—
54#875—21no—
55#874D411yescharacteristic
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74#821—24no—
75#821—22no—
76#821—22no—
77#821—24no—
78#818—24no—
79#817—22no—
80#816—24no—
81#803—21no—
82C4×C3⋊C8—21no—
83#532—31no—
84#532—31no—
85C2×C4⋊Q8—31no—
86#500—61no—
87#489—62no—
88#487—61no—
89#486—61no—
90#486—61no—
91#480—62no—
92#476—61no—
93#476—61no—
94#474—62no—
95#469—62no—
96#459—61no—
97#459—61no—
98#431C2×D311yescharacteristic
99#423C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102C4×M16C2×D311yescharacteristic
103#343—62no—
104#329—62no—
105D4⋊C8—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110C2×#67—41no—
111C22×Dic3—22no—
112#235—44no—
113#235—82no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—82no—
127C3×M16—42no—
128C3×M16—42no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148#208—22no—
149#208—41no—
150C2×C3⋊C8—41no—
151D24—82no—
152C2×#41—61no—
153C2×#41—61no—
154C22×Q8—61no—
155C2×D8—62no—
156C2×M16—21no—
157C2×M16—61no—
158C2×M16D1212yes—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#127—31no—
166#127—31no—
167#124—61no—
168#124—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#106—62no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8D1212yes—
185#103—64no—
186#102—64no—
187#102—62no—
188#102—62no—
189#102—64no—
190C22⋊C8—21no—
191#97—61no—
192#97D1212yes—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—61no—
196C23×C3#12111yescharacteristic
197C22×D3—41no—
198C22×D3—41no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209#67—82no—
210#67—82no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—82no—
215D12—42no—
216D12—42no—
217D12—41no—
218D12—41no—
219D12—41no—
220D12—41no—
221D12—42no—
222D12—81no—
223C4×D3—82no—
224C4×D3—82no—
225Dic6—42no—
226Dic6—82no—
227Dic6—82no—
228Dic6—81no—
229C3×C8—42no—
230C3×C8—42no—
231C3×C8—42no—
232C3⋊C8—42no—
233#41—124no—
234#41—242no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—121no—
238C2×Q8—121no—
239C2×Q8—242no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—121no—
247C2×D4—121no—
248C22×C4—62no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4C2×D1211yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4#23911yescharacteristic
254D8—242no—
255M16—42no—
256M16—42no—
257M16—62no—
258M16—121no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—121no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42C2×D1211yescharacteristic
276C42C2×D1211yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3#39211yescharacteristic
283C2×D3—81no—
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C2×D3—81no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4#45412yes—
293C3×C4#45511yescharacteristic
294C3×C4#45611yescharacteristic
295C3×C4#47212yes—
296Dic3—81no—
297Dic3—82no—
298Dic3—81no—
299C23—121no—
300C23—121no—
301C23#89511yescharacteristic
302Q8—122no—
303Q8—241no—
304Q8—242no—
305Q8—242no—
306D4—122no—
307D4—122no—
308D4—121no—
309D4—121no—
310D4—121no—
311D4—121no—
312D4—122no—
313D4—241no—
314D4—242no—
315D4—242no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—122no—
323C2×C4—242no—
324C2×C4—242no—
325C2×C4—242no—
326C2×C4#87411yescharacteristic
327C2×C4#88211yescharacteristic
328C2×C4#88211yescharacteristic
329C2×C4#88211yescharacteristic
330C2×C4#89511yescharacteristic
331C2×C4#89511yescharacteristic
332C8—42no—
333C8—42no—
334C8—42no—
335C8—122no—
336C2×C3—41no—
337C2×C3#200311yescharacteristic
338C2×C3#200711yescharacteristic
339C2×C3#201811yescharacteristic
340D3—161no—
341D3—161no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22#467111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—241no—
356C4—242no—
357C4—241no—
358C4#474611yescharacteristic
359C4#475211yescharacteristic
360C4#482712yes—
361C4#489712yes—
362C3#1011011yescharacteristic, complemented, Sylow
363C2—41no—
364C2—481no—
365C2—481no—
366C2#6876711yescharacteristic
367C2#6894211yescharacteristic
368C2#6900911yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1299C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1299D3
hypocenterC3#10110
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1299

Complements

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

NH ≅ G/Ntype
1. #79268C2semidirect
3. #78528C2semidirect
4. #70761C2semidirect
5. #70524C2semidirect
6. #69859C2semidirect
7. C3×#1299C2semidirect
38. #1299D3semidirect
362. C3#10110semidirect

Maximal subgroups

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

subgroupindex
#792682
#790642
#785282
#707612
#705242
#698592
C3×#12992
#101103