Finite Group Databasebeta

← #10699180

Subgroups — #10699180

Order 768.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#83213C211yesmaximal, characteristic
2C3×#1981C211yesmaximal, characteristic, complemented
3#71790C211yesmaximal, characteristic, complemented
4#70432C211yesmaximal, characteristic
5#70420C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10501—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#103—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14#5227—22no—
15#5085—21no—
16#5083—21no—
17#5083—21no—
18#5079—21no—
19C2×#805—21no—
20C2×#805—21no—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4530—21no—
25#4530—21no—
26#4486—21no—
27#4485—21no—
28#4480—21no—
29#4479—21no—
30#3206—31no—
31#1997—31no—
32#1981D311yescharacteristic, complemented
33#1980—31no—
34#1565—31no—
35#1547—31no—
36#1303—31no—
37C22×C3×Q8D412yes—
38C2×#235—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#215—22no—
50C2×#215—22no—
51#930—21no—
52#930—21no—
53C2×#208—21no—
54C2×#208D411yescharacteristic
55C2×#208D411yescharacteristic
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#869D411yescharacteristic
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×#103—44no—
64C3×#95—21no—
65#835—22no—
66#835—22no—
67#830—21no—
68#805—42no—
69#805—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#532—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79#492—61no—
80#487—61no—
81#461—61no—
82#461—61no—
83#448—62no—
84#431—31no—
85#423—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×#104—61no—
88C2×#104—61no—
89C2×#103—22no—
90C4×M16—31no—
91C4×C4⋊C4C2×D311yescharacteristic
92#343—61no—
93#343—61no—
94#335—61no—
95#334—61no—
96#332—61no—
97#331—61no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—42no—
101C2×C3×Q8—42no—
102C22×C3×C4—22no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—41no—
108#235—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—21no—
117C2×C3×C8—42no—
118C2×C3×C8—41no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#215—82no—
132#215—82no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137#208—21no—
138#208—21no—
139#208—21no—
140#208—41no—
141#208—41no—
142#208SD1612yescomplemented
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—41no—
148C2×#41—61no—
149C22×Q8#6712yes—
150C2×SD16—62no—
151C2×SD16—62no—
152C2×M16—31no—
153C2×M16—31no—
154C2×M16—61no—
155C22×C8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—31no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C42—21no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—31no—
175C4⋊C8—31no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#104—122no—
179#104—122no—
180#103—44no—
181#103—62no—
182#103—62no—
183C22⋊C8—61no—
184#97—61no—
185C4×C8—61no—
186C4×C8—61no—
187#95—21no—
188C23×C3#12111yescharacteristic
189C22×D3—41no—
190C3×Q8—44no—
191C3×Q8—42no—
192C3×Q8—42no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—82no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4#12411yescharacteristic
207C2×C3×C4C2×SD1611yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C3×C4#13711yescharacteristic
211#67—81no—
212#67—81no—
213C2×Dic3—41no—
214D12—41no—
215D12—41no—
216D12—41no—
217D12—41no—
218D12—81no—
219C4×D3—81no—
220C4×D3—81no—
221Dic6—41no—
222Dic6—41no—
223C3×C8—81no—
224C3⋊C8—22no—
225C3⋊C8—41no—
226C3⋊C8—41no—
227C3⋊C8—41no—
228C3⋊C8—81no—
229#41—121no—
230#41—121no—
231#41—121no—
232#41—121no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—121no—
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247C22×C4C2×#6711yescharacteristic
248SD16—242no—
249SD16—242no—
250M16—32no—
251M16—61no—
252M16—61no—
253M16—61no—
254M16—121no—
255M16—121no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—121no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C42—41no—
274C42—41no—
275C42#23511yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#39411yescharacteristic
282C22×C3#46011yescharacteristic
283C22×C3#46411yescharacteristic
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—82no—
291C3×C4—82no—
292C3×C4#45311yescharacteristic
293C3×C4#45511yescharacteristic
294C3×C4#46111yescharacteristic
295C3×C4#46311yescharacteristic
296C3×C4#46511yescharacteristic
297C3×C4#46811yescharacteristic
298Dic3—81no—
299C23—121no—
300C23#93911yescharacteristic
301Q8—44no—
302Q8—42no—
303Q8—42no—
304Q8—121no—
305Q8—121no—
306D4—121no—
307D4—121no—
308D4—121no—
309D4—121no—
310D4—241no—
311D4—241no—
312D4—241no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—121no—
322C2×C4—241no—
323C2×C4—241no—
324C2×C4#88311yescharacteristic
325C2×C4#88411yescharacteristic
326C2×C4#88411yescharacteristic
327C2×C4#88511yescharacteristic
328C2×C4D3×SD1611yescharacteristic
329C2×C4#91411yescharacteristic
330C2×C4#91811yescharacteristic
331C2×C4#91911yescharacteristic
332C2×C4#92911yescharacteristic
333C2×C4#93711yescharacteristic
334C8—62no—
335C8—81no—
336C8—121no—
337C8—121no—
338C8—121no—
339C8—241no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3#203711yescharacteristic
343C2×C3#204911yescharacteristic
344C2×C3#205511yescharacteristic
345D3—161no—
346C22—21no—
347C22—21no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22#495911yescharacteristic
352C22#516311yescharacteristic
353C22#518511yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—82no—
359C4—241no—
360C4#480711yescharacteristic
361C4#481211yescharacteristic
362C4#481811yescharacteristic
363C4#481911yescharacteristic
364C4#506111yescharacteristic
365C4#507711yescharacteristic
366C3#1050111yescharacteristic, complemented, Sylow
367C2—21no—
368C2—21no—
369C2—481no—
370C2#6966511yescharacteristic
371C2#7028011yescharacteristic
372C2#7035211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#394
prime radical#5477C22
hypercenter#1981D3
hypocenterC3#10501
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C3×#1981C2semidirect
3. #71790C2semidirect
5. #70420C2semidirect
7. #67620C2semidirect
32. #1981D3semidirect
142. #208SD16semidirect
366. C3#10501semidirect

Maximal subgroups

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

subgroupindex
#832132
C3×#19812
#717902
#704322
#704202
#694922
#676202
#105013