Finite Group Databasebeta

← #10767966

Subgroups — #10767966

Order 768.

1250 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×#3496C211yesmaximal, characteristic
2#83316C212yesmaximal, complemented
3#82348C211yesmaximal, characteristic, complemented
4#71846C214yesmaximal, complemented
5#71843C214yesmaximal, complemented
6#71812C212yesmaximal, complemented
7#71603C211yesmaximal, characteristic, complemented
8#18179—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)—21no—
10C3×#547—22no—
11C3×#546—24no—
12C3×D42—22no—
13C3×#537—21no—
14C3×#534C2214yes—
15C3×#532C2212yes—
16C2×C3×#121—22no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic, complemented
19#5222—21no—
20C4×#217—22no—
21#5091—22no—
22#5075—22no—
23#5040C2214yescomplemented
24#5033C2214yescomplemented
25#5018—24no—
26#4985—21no—
27#4953C2212yescomplemented
28#4947C2212yescomplemented
29#4932C2211yescharacteristic, complemented
30#4926C2212yescomplemented
31#4537C2218yescomplemented
32#4525C2214yescomplemented
33#3496D311yescharacteristic
34#3112—32no—
35#2398—31no—
36#1607—34no—
37#1601—34no—
38#1505—32no—
39#1466—31no—
40C3×H(5,𝔽2)—48no—
41C2×C3×#41—24no—
42C2×C3×#41—22no—
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—22no—
46C22×C3×D4D411yescharacteristic
47C23×C3×C4D412yes—
48C22×Dic6C2311yescharacteristic
49C3×C4⋊Q8D412yes—
50C3×#127D412yes—
51C3×#124—22no—
52C3×#123—28no—
53C3×C22⋊Q8—24no—
54C3×#121—24no—
55C3×#121—24no—
56C3×#121—24no—
57C3×#121—22no—
58C3×#121—24no—
59C3×#121C2314yes—
60C3×#120—28no—
61C3×C4×D4—22no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×C4×D4D414yes—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4C2312yes—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72C2×#214—24no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—21no—
77C4×Dic6—22no—
78C4×Dic6—22no—
79#832—24no—
80C3⋊#95—21no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95C2312yes—
84#830C2314yes—
85#810—24no—
86#807—24no—
87C4⋊(C3⋊C8)—22no—
88#803—22no—
89C2×H(5,𝔽2)—21no—
90#547—22no—
91#546—24no—
92D42—22no—
93#537—21no—
94#534C2×D314yes—
95#532C2×D312yes—
96C2×#121—22no—
97C2×C4×Q8—31no—
98C2×C4×D4C2×D311yescharacteristic
99#480—34no—
100#477—62no—
101#460—62no—
102#448—34no—
103#440—64no—
104#432—32no—
105#400—61no—
106#397—32no—
107#394—61no—
108#393—32no—
109C4×C22⋊C4—62no—
110#376—31no—
111#327—38no—
112#323—34no—
113C24×C3—24no—
114C24×C3C2×D412yes—
115C3×#41—48no—
116C3×#41—416no—
117C2×C3×Q8—22no—
118C2×C3×D4—24no—
119C2×C3×D4—28no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—416no—
126C2×C3×D4—48no—
127C2×C3×D4—48no—
128C2×C3×D4—84no—
129C2×C3×D4C2×D414yes—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—44no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2411yescharacteristic
139C22×Dic3—21no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C2×Dic6—22no—
143C2×Dic6—42no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4C2×D414yes—
148C3×C22⋊C4—28no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C42C2×D412yes—
153C3×C42C2×D412yes—
154#217—44no—
155#217—44no—
156#214—48no—
157C4⋊Dic3—22no—
158Dic3⋊C4—42no—
159C4×Dic3—42no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—24no—
163H(5,𝔽2)—48no—
164C2×#41—24no—
165C2×#41—22no—
166C22×Q8—31no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4#6711yescharacteristic
171C23×C4#6712yes—
172C2×SD16—64no—
173C4⋊Q8#6712yes—
174#127#6712yes—
175#124—22no—
176#123—28no—
177C22⋊Q8—24no—
178#121—24no—
179#121—24no—
180#121—24no—
181#121—22no—
182#121—24no—
183#121C22×D314yes—
184#120—28no—
185C4×Q8—62no—
186C4×Q8—62no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4#6714yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—24no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4C22×D312yes—
199C2×C42—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—62no—
204#103—64no—
205#102—64no—
206C22⋊C8—34no—
207#97—62no—
208#95—32no—
209#95—61no—
210#95—62no—
211#95—62no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—24no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—48no—
218C23×C3—84no—
219C23×C3—84no—
220C23×C3AGL(1,ℤ/8)14yes—
221C23×C3C22×D411yescharacteristic
222C3×Q8—44no—
223C3×D4—28no—
224C3×D4—44no—
225C3×D4—44no—
226C3×D4—416no—
227C3×D4—48no—
228C3×D4—44no—
229C3×D4—84no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—41no—
235C2×C3×C4—48no—
236C2×C3×C4—48no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—82no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12012yes—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4C22×D412yes—
244C2×Dic3—22no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252Dic6—24no—
253Dic6—42no—
254C3⋊C8—44no—
255C24—24no—
256C24C2×#6712yes—
257#41—48no—
258#41—416no—
259C2×Q8—22no—
260C2×Q8—62no—
261C2×Q8—122no—
262C2×D4—24no—
263C2×D4—28no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—416no—
270C2×D4—48no—
271C2×D4—48no—
272C2×D4—84no—
273C2×D4C2×#6714yes—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—44no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6712yes—
285C22×C4C23×D311yescharacteristic
286SD16—128no—
287C2×C8—64no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—62no—
292C4⋊C4—122no—
293C4⋊C4C2×#6714yes—
294C22⋊C4—28no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C42—122no—
301C42—121no—
302C42—121no—
303C42C2×#6712yes—
304C42C2×#6712yes—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—24no—
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—82no—
315C22×C3—168no—
316C22×C3C2×#12011yescharacteristic
317C22×C3C2×AGL(1,ℤ/8)12yes—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—44no—
322C3×C4—42no—
323C3×C4#45512yes—
324Dic3—42no—
325Dic3—82no—
326Dic3—81no—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—42no—
331C23—42no—
332C23—48no—
333C23—84no—
334C23—84no—
335C23#93214yes—
336C23C22×#6711yescharacteristic
337Q8—44no—
338Q8—64no—
339Q8—122no—
340D4—28no—
341D4—44no—
342D4—44no—
343D4—416no—
344D4—48no—
345D4—44no—
346D4—84no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—41no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—62no—
357C2×C4—82no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—121no—
365C2×C4#95311yescharacteristic
366C2×C4#95312yes—
367C2×C4#95311yescharacteristic
368C2×C4C22×#6712yes—
369C8—124no—
370C2×C3—22no—
371C2×C3—44no—
372C2×C3—84no—
373C2×C3#300311yescharacteristic
374C2×C3C2×#45511yescharacteristic
375C2×C3#302511yescharacteristic
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—44no—
383C22—44no—
384C22—44no—
385C22—82no—
386C22—168no—
387C22C2×#93212yes—
388C22C2×#95311yescharacteristic
389C4—22no—
390C4—21no—
391C4—22no—
392C4—44no—
393C4—42no—
394C4—122no—
395C4—242no—
396C4—241no—
397C4#524212yes—
398C3#1817911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—44no—
401C2—84no—
402C2#8403811yescharacteristic
403C2C2×#524211yescharacteristic
404C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#18179
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
2. #83316C2semidirect
3. #82348C2semidirect
4. #71846C2semidirect
5. #71843C2semidirect
6. #71812C2semidirect
7. #71603C2semidirect
18. C2×C4×Dic6C22semidirect
23. #5040C22semidirect
24. #5033C22semidirect
27. #4953C22semidirect
28. #4947C22semidirect
29. #4932C22semidirect
30. #4926C22semidirect
31. #4537C22semidirect
32. #4525C22semidirect
398. C3#18179semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
#833162
#823482
#718462
#718432
#718122
#716032
#181793