Finite Group Databasebeta

← #10696629

Subgroups — #10696629

Order 768.

568 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#82980C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic
5#70836C211yesmaximal, characteristic, complemented
6#70822C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10102—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#944—21no—
11C2×#936—21no—
12#5470C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×C4×M16C2211yescharacteristic
16#5228—21no—
17#5218—21no—
18#5121—22no—
19#5060—21no—
20#5055—21no—
21#5054—22no—
22#4949—22no—
23#4949—22no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26#4702—22no—
27#4682—21no—
28#4680—21no—
29C3×#329—22no—
30#4486—22no—
31#4467—22no—
32#3215—31no—
33#3165—31no—
34#2658—31no—
35#1556—31no—
36#1527D311yescharacteristic
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#217—24no—
51#944—44no—
52C2×#216—22no—
53#936—44no—
54C2×#214—22no—
55#926—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58#924—21no—
59#924—21no—
60C2×Dic3⋊C4—24no—
61C2×C4×Dic3—21no—
62#870—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3×C4×C8—21no—
76#835—22no—
77#832—22no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81#818—24no—
82#817—24no—
83C4⋊(C3⋊C8)D412yes—
84#803—21no—
85#533—31no—
86C2×C4⋊Q8—31no—
87C2×C4⋊Q8—61no—
88C2×#124—61no—
89C2×#124C2×D311yescharacteristic
90#501—62no—
91#500—61no—
92#498—61no—
93#497—61no—
94#489—61no—
95#464—62no—
96#451—61no—
97#450—61no—
98#432—31no—
99#432C2×D311yescharacteristic
100#428—62no—
101C4×M16C2×D311yescharacteristic
102#399—62no—
103#399—62no—
104#376—31no—
105#332—62no—
106#329—22no—
107#328—62no—
108C24×C3—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—21no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C3×M16—24no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—48no—
139#217—41no—
140#216—82no—
141#214—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—48no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—24no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
156C22×Q8—61no—
157C22×Q8D1211yescharacteristic
158C22×D4D1211yescharacteristic
159C2×Q16—62no—
160C2×SD16—62no—
161C2×M16—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—124no—
169#125—61no—
170#124—21no—
171#124—21no—
172#124—21no—
173#124—21no—
174#124—124no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179#117—61no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—64no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—64no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—64no—
190C4⋊C8—21no—
191C4⋊C8—32no—
192#103—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196#97—21no—
197#97—61no—
198#97#6712yes—
199C4×C8—21no—
200#95—32no—
201#95—61no—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C3×D4—42no—
210C3×D4—81no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×Dic3—24no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230Dic6—42no—
231Dic6—82no—
232Dic6—82no—
233Dic6—81no—
234C3×C8—24no—
235C3×C8—81no—
236C3⋊C8—42no—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—42no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—122no—
244C2×Q8—122no—
245C2×Q8—121no—
246C2×Q8—121no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4C2×D1211yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256Q16—242no—
257SD16—242no—
258M16—24no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—62no—
262C4⋊C4—61no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—128no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—128no—
278C22⋊C4—121no—
279C42—64no—
280C42—121no—
281C42#23511yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—84no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C22×C3#45612yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4#45312yes—
300Dic3—44no—
301Dic3—81no—
302Dic3—82no—
303Dic3—81no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—42no—
308C23#88211yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—122no—
312Q8—241no—
313Q8—242no—
314Q8—242no—
315D4—42no—
316D4—81no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—64no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—122no—
330C2×C4#88211yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#93011yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#94011yescharacteristic
336C8—24no—
337C8—81no—
338C8—122no—
339C2×C3—22no—
340C2×C3—161no—
341C2×C3#200211yescharacteristic
342C2×C3#200511yescharacteristic
343C2×C3#200911yescharacteristic
344C22—21no—
345C22—21no—
346C22—84no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#482612yes—
351C22#496011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—124no—
359C4—241no—
360C4—242no—
361C4—241no—
362C4#474712yes—
363C3#1010211yescharacteristic, complemented, Sylow
364C2—22no—
365C2—161no—
366C2#7008211yescharacteristic
367C2#7038911yescharacteristic
368C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1527D3
hypocenterC3#10102
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
5. #70836C2semidirect
6. #70822C2semidirect
7. #67665C2semidirect
363. C3#10102semidirect

Maximal subgroups

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

subgroupindex
#829802
#829492
#785062
C3×#15272
#708362
#708222
#676652
#101023