Finite Group Databasebeta

← #10697432

Subgroups — #10697432

Order 768.

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

normal subgroups only

This group has 542 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×#3208C211yesmaximal, characteristic
2#83340C211yesmaximal, characteristic, complemented
3#83169C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71765C211yesmaximal, characteristic
6#70793C211yesmaximal, characteristic, complemented
7#67741C211yesmaximal, characteristic, complemented
8#10220—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#492—21no—
13C3×#486—21no—
14C3×#460—22no—
15C3×#450—22no—
16C3×#431C2211yescharacteristic
17#5218—22no—
18C2×#832—22no—
19#5081—21no—
20#5081—21no—
21#5076—21no—
22#5075—22no—
23#5070—21no—
24#5064—21no—
25#5064—21no—
26#4998—22no—
27#4936C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4538—22no—
31#4467—22no—
32#3208—31no—
33#3208D311yescharacteristic
34#3162—31no—
35#1961—31no—
36#1518—31no—
37#1518—31no—
38#1281—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×C3×SD16—24no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127D412yes—
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#214—24no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59#924—21no—
60#924—21no—
61C22×C3⋊C8D412yes—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×#102—24no—
69C3×C22⋊C8D412yes—
70C3×C4×C8—21no—
71#832—24no—
72#832—44no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#830D412yes—
76#808—22no—
77#808—22no—
78#807—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#533C2×D311yescharacteristic
86C2×#127C2×D311yescharacteristic
87#501—61no—
88#501—61no—
89#501—61no—
90#501—61no—
91#492—21no—
92#492—61no—
93#486—21no—
94#486—61no—
95#460—22no—
96#460—62no—
97#450—22no—
98#450—62no—
99#431—31no—
100#431C2×D311yescharacteristic
101#427—62no—
102C2×#102—62no—
103C2×#97—31no—
104#382—31no—
105#328—62no—
106#328—62no—
107C24×C3—41no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—82no—
119C2×C3×D4—82no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C3×SD16—84no—
127C2×C3×C8—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#214—84no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—42no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8SD1614yescomplemented
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C2×SD16—24no—
158C2×SD16—64no—
159C22×C8—32no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164#127—21no—
165#127—21no—
166#127—21no—
167#127#6712yes—
168#125—21no—
169#125—61no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178#117—21no—
179#117—61no—
180C2×C4⋊C4—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#107—62no—
184#106—62no—
185#106—62no—
186C4⋊C8—21no—
187C4⋊C8—61no—
188#102—24no—
189#102—64no—
190#102—124no—
191C22⋊C8—32no—
192C22⋊C8#6712yes—
193#97—61no—
194#97—61no—
195C4×C8—21no—
196C4×C8—61no—
197#95—61no—
198#95—61no—
199C23×C3—42no—
200C23×C3—82no—
201C23×C3—82no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3#12711yescharacteristic
205C3×Q8—42no—
206C3×Q8—81no—
207C3×D4—44no—
208C3×D4—44no—
209C3×D4—44no—
210C3×D4—82no—
211C3×D4—82no—
212C3×D4—82no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4C2×SD1612yes—
224C2×C3×C4#13712yes—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230Dic6—42no—
231Dic6—81no—
232C3×C8—42no—
233C3⋊C8—24no—
234C3⋊C8—42no—
235C24—41no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C22×C4—121no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254SD16—84no—
255SD16—244no—
256C2×C8—22no—
257C2×C8—34no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—122no—
274C22⋊C4—41no—
275C22⋊C4—121no—
276C42—41no—
277C42—121no—
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C42C2×#6711yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—162no—
288C22×C3—162no—
289C22×C3#39211yescharacteristic
290C22×C3#49211yescharacteristic
291C22×C3#49711yescharacteristic
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4#44812yes—
299C3×C4#46112yes—
300Dic3—81no—
301Dic3—81no—
302Dic3—161no—
303C23—42no—
304C23—82no—
305C23—82no—
306C23—81no—
307C23—81no—
308C23#94011yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—122no—
312Q8—241no—
313D4—44no—
314D4—44no—
315D4—44no—
316D4—82no—
317D4—82no—
318D4—82no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4D3×SD1612yes—
329C2×C4#91512yes—
330C2×C4#93711yescharacteristic
331C2×C4#93711yescharacteristic
332C2×C4#93811yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#95311yescharacteristic
336C8—42no—
337C8—64no—
338C8—122no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—162no—
342C2×C3#200811yescharacteristic
343C2×C3#200811yescharacteristic
344C2×C3#201311yescharacteristic
345C22—21no—
346C22—21no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—162no—
351C22—162no—
352C22#516711yescharacteristic
353C22#517511yescharacteristic
354C22#522311yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—81no—
361C4—241no—
362C4—241no—
363C4—481no—
364C4#516212yes—
365C4#517012yes—
366C3#1022011yescharacteristic, complemented, Sylow
367C2—21no—
368C2—21no—
369C2—162no—
370C2#7185811yescharacteristic
371C2#7205711yescharacteristic
372C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3208C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#3208D3
hypocenterC3#10220
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
2. #83340C2semidirect
4. #71838C2semidirect
6. #70793C2semidirect
7. #67741C2semidirect
153. C2×C3⋊C8SD16semidirect
366. C3#10220semidirect

Maximal subgroups

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

subgroupindex
C3×#32082
#833402
#831692
#718382
#717652
#707932
#677412
#102203