Finite Group Databasebeta

← #10696569

Subgroups — #10696569

Order 768.

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

normal subgroups only

This group has 590 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×#3213C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#82980C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70823C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10094—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#944—21no—
12C2×#936—21no—
13C2×#869C2211yescharacteristic
14C3×#496—21no—
15C3×#489—21no—
16C3×#463—22no—
17C3×#448—21no—
18C3×#447—21no—
19C3×#432C2211yescharacteristic
20#5232—22no—
21#5079—21no—
22#5070—21no—
23#5055—22no—
24#4949—22no—
25#4949—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4540—22no—
30#4537—22no—
31#4482—22no—
32#3213D311yescharacteristic
33#3141—31no—
34#2658—31no—
35#1556—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×Dic6—21no—
43C2×C3×SD16—22no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—24no—
60#944—44no—
61#936—44no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D412yes—
64C2×Dic3⋊C4—24no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8D412yes—
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76#835—24no—
77#832—24no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×C4⋊Q8—61no—
88C2×#124—61no—
89C2×#124C2×D311yescharacteristic
90#496—21no—
91#489—21no—
92#489—62no—
93#487—61no—
94#486—61no—
95#463—22no—
96#448—21no—
97#447—21no—
98#432—31no—
99#432C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102#399—62no—
103#399—62no—
104#376—31no—
105#332—62no—
106#329—62no—
107#327—62no—
108C24×C3—21no—
109C3×#41—44no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—22no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C3×SD16—82no—
133C3×D8—82no—
134C2×C3×C8—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—48no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152Dic3⋊C4—48no—
153C4×Dic3—24no—
154#208—42no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×#41—21no—
158C22×Q8—61no—
159C22×Q8#6711yescharacteristic
160C22×D4#6711yescharacteristic
161C2×SD16—22no—
162C2×D8—22no—
163C2×M16—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—124no—
169C4⋊Q8#6711yescharacteristic
170#127#6711yescharacteristic
171#124—21no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176#124—124no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—64no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—22no—
192#103—64no—
193#102—22no—
194#102—64no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—61no—
200C4×C8—61no—
201#95—32no—
202#95—61no—
203C23×C3—42no—
204C23×C3—42no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3#12011yescharacteristic
209C3×Q8—42no—
210C3×Q8—42no—
211C3×Q8—81no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—82no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—82no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×Dic3—24no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237Dic6—82no—
238Dic6—82no—
239C3×C8—42no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—21no—
243#41—44no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—122no—
249C2×Q8—122no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—41no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—82no—
267D8—82no—
268M16—122no—
269C2×C8—22no—
270C2×C8—62no—
271C2×C8—121no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—128no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—128no—
286C42—64no—
287C42D3×D411yescharacteristic
288C42#23711yescharacteristic
289C42C2×#6711yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—84no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3#39211yescharacteristic
301C22×C3#45412yes—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4#45312yes—
310Dic3—44no—
311Dic3—82no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—41no—
316C23—41no—
317C23#93711yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—81no—
321Q8—242no—
322Q8—242no—
323D4—42no—
324D4—42no—
325D4—42no—
326D4—81no—
327D4—81no—
328D4—82no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—64no—
336C2×C4—82no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4#93711yescharacteristic
343C2×C4#93811yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#94011yescharacteristic
347C2×C4#95311yescharacteristic
348C8—42no—
349C8—122no—
350C8—241no—
351C2×C3—22no—
352C2×C3—161no—
353C2×C3—161no—
354C2×C3#200211yescharacteristic
355C2×C3#200311yescharacteristic
356C2×C3#200911yescharacteristic
357C22—21no—
358C22—21no—
359C22—84no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22#520112yes—
367C22#522311yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—81no—
374C4—81no—
375C4—124no—
376C4—242no—
377C4#524112yes—
378C3#1009411yescharacteristic, complemented, Sylow
379C2—22no—
380C2—161no—
381C2—161no—
382C2#7185611yescharacteristic
383C2#7209611yescharacteristic
384C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3213C2
socleC22×C3#392
hypercenter#3213D3
hypocenterC3#10094
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3213

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #82980C2semidirect
4. #71812C2semidirect
5. #71718C2semidirect
6. #70823C2semidirect
7. #67748C2semidirect
378. C3#10094semidirect

Maximal subgroups

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

subgroupindex
C3×#32132
#832832
#829802
#718122
#717182
#708232
#677482
#100943