Finite Group Databasebeta

← #10697345

Subgroups — #10697345

Order 768.

512 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3226C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#83198C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic, complemented
6#70792C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#10205—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#500—21no—
13C3×#486—21no—
14C3×#480—22no—
15C3×#474—22no—
16C3×#431C2211yescharacteristic
17C2×#837—22no—
18#5225—22no—
19#5094—21no—
20#5082—22no—
21#5079—21no—
22#5079—21no—
23#5075—21no—
24#5041—22no—
25#5019—21no—
26#5008—21no—
27C4×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4529—22no—
31#4467—22no—
32#3246—31no—
33#3226D311yescharacteristic
34#3139—31no—
35#1988—31no—
36#1980—31no—
37#1503—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#214—22no—
56#926—21no—
57C2×C4⋊Dic3—21no—
58#924—21no—
59#924—21no—
60C2×C4×Dic3—21no—
61C2×#208D412yes—
62C22×C3⋊C8D412yes—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#107—24no—
69C3×C4⋊C8—21no—
70C3×#102—24no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#837—44no—
74#835—24no—
75#835—24no—
76#832—22no—
77C3⋊#95—21no—
78#810—22no—
79#808—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87#500—21no—
88#491—62no—
89#487—61no—
90#487—61no—
91#486—21no—
92#480—22no—
93#476—61no—
94#475—61no—
95#474—22no—
96#465—62no—
97#461—61no—
98#460—61no—
99#431C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#104—62no—
102#419—62no—
103C4×M16—31no—
104C4×C4⋊C4—31no—
105#340—62no—
106#328—62no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140#214—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C4×Dic3—42no—
151#208—22no—
152#208—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—42no—
156C2×#41—21no—
157C22×Q8—21no—
158C2×SD16—62no—
159C2×M16—32no—
160C22×C8—32no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8#6712yes—
167C4⋊Q8#6711yescharacteristic
168#127#6711yescharacteristic
169#125—61no—
170#124—21no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—21no—
174#121—21no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×D4—21no—
178C4×D4—21no—
179#117—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#107—24no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#104—124no—
191#103—62no—
192#103—64no—
193#103—64no—
194#102—24no—
195#102—62no—
196C22⋊C8#6712yes—
197#97—61no—
198C4×C8—21no—
199C4×C8—61no—
200#95—61no—
201C23×C3—41no—
202C23×C3#12711yescharacteristic
203C3×Q8—44no—
204C3×Q8—42no—
205C3×Q8—82no—
206C3×D4—42no—
207C3×D4—41no—
208C3×D4—41no—
209C3×D4—42no—
210C3×D4—81no—
211C3×D4—81no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—82no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#13512yes—
227C2×C3×C4AGL(1,ℤ/8)12yes—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233Dic6—41no—
234Dic6—41no—
235Dic6—81no—
236C3×C8—42no—
237C3⋊C8—42no—
238C3⋊C8—42no—
239#41—42no—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—121no—
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4C2×#6711yescharacteristic
258SD16—242no—
259M16—62no—
260M16—62no—
261C2×C8—22no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—122no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—121no—
282C42—121no—
283C42—122no—
284C42D3×D411yescharacteristic
285C42#23711yescharacteristic
286C42C2×#6711yescharacteristic
287C42C2×#6711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#49511yescharacteristic
295C22×C3#49611yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4—82no—
301C3×C4#44912yes—
302C3×C4#45511yescharacteristic
303C3×C4#45611yescharacteristic
304C3×C4#46312yes—
305Dic3—82no—
306Dic3—81no—
307Dic3—81no—
308C23—41no—
309C23#94011yescharacteristic
310Q8—42no—
311Q8—44no—
312Q8—82no—
313Q8—121no—
314Q8—121no—
315Q8—241no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4—81no—
321D4—81no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—241no—
334C2×C4—241no—
335C2×C4#91112yes—
336C2×C4#91212yes—
337C2×C4#93711yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#93811yescharacteristic
340C2×C4#93911yescharacteristic
341C2×C4#93911yescharacteristic
342C2×C4#95311yescharacteristic
343C8—42no—
344C8—122no—
345C8—122no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—161no—
349C2×C3#200711yescharacteristic
350C2×C3#200911yescharacteristic
351C2×C3#201211yescharacteristic
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#515411yescharacteristic
358C22#515511yescharacteristic
359C22#522311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—82no—
365C4—242no—
366C4—241no—
367C4—241no—
368C4#515312yes—
369C4#515812yes—
370C4#520211yescharacteristic
371C4#520511yescharacteristic
372C3#1020511yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—161no—
376C2#7186911yescharacteristic
377C2#7204511yescharacteristic
378C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3226C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3226D3
hypocenterC3#10205
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
3. #83198C2semidirect
5. #71780C2semidirect
6. #70792C2semidirect
7. #67744C2semidirect
372. C3#10205semidirect

Maximal subgroups

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

subgroupindex
C3×#32262
#833202
#831982
#718182
#717802
#707922
#677442
#102053