Finite Group Databasebeta

← #10696597

Subgroups — #10696597

Order 768.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3215C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic, complemented
3#82978C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic
6#70824C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10098—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#497—21no—
13C3×#489—21no—
14C3×#464—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#432C2211yescharacteristic
18#5229—21no—
19#5221—21no—
20#5093—21no—
21#5093—21no—
22#5069—22no—
23#5004—22no—
24#4953—22no—
25#4947—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4540—22no—
30#4538—22no—
31#4481—22no—
32#3215D311yescharacteristic
33#3139—31no—
34#2640—31no—
35#1558—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6D412yes—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68C3×C4⋊C8D412yes—
69C3×#103—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3⋊#95—24no—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#810—24no—
78#810—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#124C2×D311yescharacteristic
86#497—21no—
87#491—62no—
88#489—21no—
89#487—61no—
90#487—61no—
91#464—22no—
92#451—21no—
93#450—21no—
94#432—31no—
95#432C2×D311yescharacteristic
96#419—62no—
97C4×M16—31no—
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#333—62no—
104#329—62no—
105#328—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—22no—
120C22×Dic3—22no—
121C2×Dic6—22no—
122C2×Dic6—22no—
123C2×Dic6—44no—
124C3×Q16—82no—
125C3×SD16—82no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—84no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4×Dic3—42no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C22×Q8—32no—
153C22×Q8#6711yescharacteristic
154C22×D4#6711yescharacteristic
155C2×Q16—22no—
156C2×SD16—22no—
157C2×M16—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164#125—21no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—64no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181C4⋊C8—61no—
182C4⋊C8#6712yes—
183#103—22no—
184#103—64no—
185#103—64no—
186#102—22no—
187C22⋊C8—21no—
188C22⋊C8—61no—
189#97—21no—
190#97—32no—
191#97—61no—
192C4×C8—61no—
193#95—32no—
194#95—64no—
195#95—61no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3#12011yescharacteristic
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×Q8—81no—
205C3×D4—42no—
206C3×D4—81no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—44no—
225Dic6—44no—
226Dic6—82no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—42no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—124no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4C2×#6711yescharacteristic
248Q16—82no—
249SD16—82no—
250M16—122no—
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—244no—
270C42—41no—
271C42—122no—
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—84no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3#39211yescharacteristic
283C22×C3#45512yes—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4#45512yes—
293Dic3—82no—
294Dic3—82no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23#93711yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304Q8—124no—
305Q8—242no—
306D4—42no—
307D4—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—124no—
319C2×C4—122no—
320C2×C4#93711yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#93911yescharacteristic
324C2×C4#94011yescharacteristic
325C2×C4#95311yescharacteristic
326C8—42no—
327C8—122no—
328C8—241no—
329C2×C3—22no—
330C2×C3—161no—
331C2×C3#200411yescharacteristic
332C2×C3#200411yescharacteristic
333C2×C3#200911yescharacteristic
334C22—21no—
335C22—21no—
336C22—84no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#520212yes—
341C22#522311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—242no—
351C4—242no—
352C4#524212yes—
353C3#1009811yescharacteristic, complemented, Sylow
354C2—22no—
355C2—161no—
356C2#7185711yescharacteristic
357C2#7210611yescharacteristic
358C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3215D3
hypocenterC3#10098
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #83337C2semidirect
4. #71836C2semidirect
6. #70824C2semidirect
7. #67748C2semidirect
353. C3#10098semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833372
#829782
#718362
#717182
#708242
#677482
#100983