Finite Group Databasebeta

← #10697317

Subgroups — #10697317

Order 768.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3227C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#83207C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic, complemented
6#70792C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#10201—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#498—21no—
13C3×#486—21no—
14C3×#481—22no—
15C3×#476—22no—
16C3×#431C2211yescharacteristic
17C2×#837—22no—
18C2×#835—22no—
19#5094—21no—
20#5081—22no—
21#5080—21no—
22#5078—21no—
23#5075—21no—
24#5041—22no—
25#5019—21no—
26#5008—21no—
27#4964—22no—
28C4×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4467—22no—
32#3246—31no—
33#3227D311yescharacteristic
34#3165—31no—
35#1981—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×#106—24no—
69C3×C4⋊C8—21no—
70C3×#102—24no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#837—44no—
74#835—44no—
75#832—22no—
76C3⋊#95—21no—
77#810—22no—
78#808—24no—
79#808—22no—
80#807—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8C2×D311yescharacteristic
88#501—62no—
89#500—61no—
90#498—21no—
91#498—61no—
92#486—21no—
93#481—22no—
94#476—22no—
95#476—61no—
96#475—61no—
97#465—62no—
98#461—61no—
99#460—61no—
100#431C2×D311yescharacteristic
101#428—62no—
102C2×C4⋊C8—31no—
103C2×#104—62no—
104C2×#103—62no—
105C4×M16—31no—
106C4×C4⋊C4—31no—
107#328—62no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—41no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—41no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×C3×C8—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#214—82no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—42no—
152#208—22no—
153#208SD1614yescomplemented
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C2×#41—21no—
158C22×Q8—21no—
159C2×SD16—62no—
160C2×M16—32no—
161C22×C8—32no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8#6712yes—
168C4⋊Q8#6711yescharacteristic
169#127#6711yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—24no—
188#106—64no—
189#106—62no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#104—124no—
194#103—62no—
195#103—124no—
196#102—24no—
197#102—62no—
198C22⋊C8#6712yes—
199#97—61no—
200C4×C8—21no—
201C4×C8—61no—
202#95—61no—
203C23×C3—41no—
204C23×C3#12711yescharacteristic
205C3×Q8—44no—
206C3×Q8—42no—
207C3×Q8—81no—
208C3×Q8—81no—
209C3×D4—42no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—42no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—82no—
221C2×C3×C4—82no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4C2×SD1612yes—
229C2×C3×C4#13712yes—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235Dic6—41no—
236Dic6—41no—
237Dic6—81no—
238C3×C8—42no—
239C3⋊C8—24no—
240C3⋊C8—42no—
241#41—42no—
242#41—42no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—121no—
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—242no—
261M16—34no—
262M16—62no—
263C2×C8—22no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—122no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—121no—
284C42—121no—
285C42—122no—
286C42D3×D411yescharacteristic
287C42#23711yescharacteristic
288C42C2×#6711yescharacteristic
289C42C2×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#39211yescharacteristic
296C22×C3#49211yescharacteristic
297C22×C3#49711yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—81no—
303C3×C4#44812yes—
304C3×C4#45311yescharacteristic
305C3×C4#45411yescharacteristic
306C3×C4#46112yes—
307Dic3—82no—
308Dic3—81no—
309Dic3—81no—
310C23—41no—
311C23#94011yescharacteristic
312Q8—44no—
313Q8—42no—
314Q8—81no—
315Q8—81no—
316Q8—121no—
317Q8—121no—
318Q8—241no—
319D4—42no—
320D4—42no—
321D4—41no—
322D4—41no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—122no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—241no—
336C2×C4—241no—
337C2×C4D3×SD1612yes—
338C2×C4#91512yes—
339C2×C4#93711yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#93811yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4#93911yescharacteristic
344C2×C4#95311yescharacteristic
345C8—42no—
346C8—64no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—161no—
351C2×C3#200611yescharacteristic
352C2×C3#200911yescharacteristic
353C2×C3#201311yescharacteristic
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#516711yescharacteristic
360C22#517511yescharacteristic
361C22#522311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—82no—
367C4—242no—
368C4—241no—
369C4—241no—
370C4#516212yes—
371C4#517012yes—
372C4#519811yescharacteristic
373C4#520111yescharacteristic
374C3#1020111yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—161no—
378C2#7186911yescharacteristic
379C2#7205711yescharacteristic
380C2#7209811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3227C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3227D3
hypocenterC3#10201
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
3. #83207C2semidirect
5. #71778C2semidirect
6. #70792C2semidirect
7. #67744C2semidirect
153. #208SD16semidirect
374. C3#10201semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#833202
#832072
#718182
#717782
#707922
#677442
#102013