Finite Group Databasebeta

← #10691460

Subgroups — #10691460

Order 768.

604 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 115 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#79309C211yesmaximal, characteristic
2C4×#887C211yesmaximal, characteristic, complemented
3#70512C211yesmaximal, characteristic
4#70500C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic
6C3×#1303C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9234—31nomaximal, Sylow
9C2×#887—21no—
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#825—21no—
16C2×#825—21no—
17Dic3×C4⋊C4—21no—
18C4×Dic3⋊C4—21no—
19C42×Dic3—21no—
20C42×Dic3C2211yescharacteristic
21#4925C2211yescharacteristic
22#4851C412yescomplemented
23#4848C412yescomplemented
24#4738—21no—
25#4735—21no—
26C4×#204—22no—
27#4649—22no—
28#4484—22no—
29#4451—22no—
30#3103—31no—
31C42×Q8—31no—
32#1962—31no—
33#1502—31no—
34#1487—31no—
35#1303D311yescharacteristic, complemented
36#1276—31no—
37C2×#235—21no—
38C22×Dic6D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—22no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#204—22no—
56#887—28no—
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6C2×C412yes—
65C4×Dic6D412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95D412yes—
72C3⋊#95—22no—
73#830—21no—
74#825—42no—
75#825—42no—
76#821—22no—
77#817—22no—
78#816—22no—
79C4⋊(C3⋊C8)—21no—
80C4×C3⋊C8—21no—
81#532—31no—
82C2×C4×Q8—31no—
83C2×C4×Q8—61no—
84#481—61no—
85#475—32no—
86#460—32no—
87#448—61no—
88C4×SD16—62no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92C2×#104—61no—
93C2×#104—61no—
94C4×C4⋊C4—61no—
95C4×C4⋊C4—61no—
96C4×C4⋊C4—62no—
97C4×C4⋊C4C2×D311yescharacteristic
98C43—31no—
99C43—61no—
100#327—62no—
101Q8⋊C8—62no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C2×#67—41no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111#235—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—24no—
121C2×Dic6—21no—
122C2×Dic6C22⋊C412yes—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—24no—
128C3×C42—41no—
129C3×C42—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3C22⋊C412yes—
140Dic3⋊C4—28no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143C4×Dic3—28no—
144C4×Dic3—24no—
145C4×Dic3—22no—
146C4×Dic3—22no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—42no—
151C4×Dic3—44no—
152C4×Dic3SD1612yescomplemented
153C2×C3⋊C8—41no—
154#204—44no—
155C2×#41—61no—
156C22×Q8—31no—
157C2×SD16—62no—
158C2×M16—21no—
159C4⋊Q8—31no—
160#127—31no—
161#124—61no—
162#121—61no—
163#121—61no—
164C4×Q8—32no—
165C4×Q8—32no—
166C4×Q8—68no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—62no—
174C2×C4⋊C4—61no—
175C2×C42—21no—
176C2×C42—31no—
177C2×C42—31no—
178C2×C42—62no—
179C2×C42—61no—
180C2×C42—61no—
181C2×C42—62no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—61no—
188C4⋊C8C4×D312yes—
189#104—122no—
190#104—122no—
191#103—62no—
192#102—62no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C4×C8—21no—
196C4×C8—61no—
197#95—62no—
198#95D1212yes—
199C23×C3#12011yescharacteristic
200C22×D3—41no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4C2×SD1611yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
215C2×C3×C4#13711yescharacteristic
216#67—81no—
217#67—81no—
218C2×Dic3—22no—
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229D12—42no—
230D12—41no—
231D12—41no—
232D12—81no—
233C4×D3—81no—
234C4×D3—81no—
235Dic6—24no—
236Dic6—22no—
237Dic6—41no—
238Dic6—41no—
239Dic6#10414yescomplemented
240C3×C8—42no—
241C3×C8—81no—
242C3⋊C8—81no—
243#41—121no—
244#41—121no—
245#41—121no—
246#41—121no—
247C2×Q8—32no—
248C2×Q8—64no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C22×C4—22no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4C2×D1211yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264SD16—124no—
265M16—41no—
266M16—41no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—121no—
270C4⋊C4—24no—
271C4⋊C4—32no—
272C4⋊C4—68no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—32no—
281C42—41no—
282C42—41no—
283C42—64no—
284C42—68no—
285C42—62no—
286C42—62no—
287C42—61no—
288C42—61no—
289C42—61no—
290C42—122no—
291C42—124no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42#23911yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#44811yescharacteristic
300C22×C3#44911yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—44no—
308C3×C4C2×#10412yes—
309C3×C4C4×SD1611yescharacteristic
310C3×C4#44011yescharacteristic
311C3×C4#46011yescharacteristic
312C3×C4#46411yescharacteristic
313Dic3—24no—
314Dic3—24no—
315Dic3—42no—
316Dic3—42no—
317Dic3—81no—
318C23—121no—
319C23#88211yescharacteristic
320Q8—34no—
321Q8—64no—
322Q8—62no—
323Q8—121no—
324Q8—121no—
325D4—121no—
326D4—122no—
327D4—121no—
328D4—241no—
329D4—241no—
330D4—241no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—62no—
337C2×C4—64no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4C4×D1211yescharacteristic
350C2×C4D3×C22⋊C411yescharacteristic
351C2×C4#88411yescharacteristic
352C2×C4#89311yescharacteristic
353C2×C4#89411yescharacteristic
354C2×C4#89511yescharacteristic
355C2×C4#91111yescharacteristic
356C2×C4D3×SD1611yescharacteristic
357C2×C4#91611yescharacteristic
358C2×C4#91811yescharacteristic
359C8—42no—
360C8—81no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#188011yescharacteristic
365C2×C3#189611yescharacteristic
366C2×C3#190611yescharacteristic
367D3—161no—
368C22—21no—
369C22—21no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22#466511yescharacteristic
374C22#478111yescharacteristic
375C22#480611yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—64no—
381C4—64no—
382C4—122no—
383C4—122no—
384C4—241no—
385C4#476211yescharacteristic
386C4#479811yescharacteristic
387C4D3×#10412yes—
388C4#485211yescharacteristic
389C4#488911yescharacteristic
390C3#923411yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—481no—
394C2#6875611yescharacteristic
395C2#6890911yescharacteristic
396C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9234
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
2. C4×#887C2semidirect
4. #70500C2semidirect
6. C3×#1303C2semidirect
7. #67736C2semidirect
22. #4851C4semidirect
23. #4848C4semidirect
35. #1303D3semidirect
152. C4×Dic3SD16semidirect
239. Dic6#104semidirect
390. C3#9234semidirect

Maximal subgroups

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

subgroupindex
#793092
C4×#8872
#705122
#705002
#694692
C3×#13032
#677362
#92343