Finite Group Databasebeta

← #10696589

Subgroups — #10696589

Order 768.

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

normal subgroups only

This group has 542 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×#3138C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic, complemented
3#83180C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71760C211yesmaximal, characteristic
6#69863C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10097—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#490—22no—
13C3×#486—21no—
14C3×#486—21no—
15C3×#418—22no—
16C3×C4×M16C2211yescharacteristic
17#5222—22no—
18#5222—22no—
19#5221—21no—
20#5221—21no—
21#5082—21no—
22#5081—21no—
23#5044—21no—
24#5043—21no—
25#5022—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4544—22no—
30#4538—22no—
31#4465—22no—
32#3236—31no—
33#3138D311yescharacteristic
34#2641—31no—
35#1555—31no—
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
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×#217—24no—
57C2×#217—24no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67C3×#102—24no—
68C3×#102—24no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71#835—22no—
72#835—22no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#533C2×D311yescharacteristic
84C2×#127C2×D311yescharacteristic
85C2×#125—31no—
86#501—61no—
87#491—61no—
88#490—22no—
89#486—21no—
90#486—21no—
91#484—61no—
92#483—61no—
93#479—62no—
94#432—31no—
95#432—31no—
96#418—22no—
97C4×M16C2×D311yescharacteristic
98#394—62no—
99#394—62no—
100#390—61no—
101#390—61no—
102#376—31no—
103#331—62no—
104#330—62no—
105#328—62no—
106C24×C3—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—44no—
112C2×C3×D4—44no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—82no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C3×M16—42no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—84no—
139#217—84no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4#6712yes—
154C2×M16—21no—
155C4⋊Q8#6711yescharacteristic
156C4⋊Q8#6711yescharacteristic
157#127—21no—
158#127—21no—
159#127—21no—
160#127—21no—
161#125—21no—
162#125—61no—
163#125—61no—
164#125—61no—
165#125—61no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170#117—21no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C22⋊C4—64no—
176C2×C22⋊C4—64no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179#107—62no—
180#106—62no—
181C4⋊C8—32no—
182C4⋊C8—61no—
183#103—62no—
184#103—62no—
185#102—24no—
186#102—24no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#97—61no—
190#97—61no—
191#97#6712yes—
192C4×C8—21no—
193#95—32no—
194#95—61no—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3—42no—
198C23×C3—44no—
199C23×C3#12011yescharacteristic
200C3×Q8—42no—
201C3×Q8—81no—
202C3×D4—44no—
203C3×D4—82no—
204C3×D4—82no—
205C3×D4—82no—
206C3×D4—82no—
207C3×D4—82no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C24—22no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—82no—
241C22×C4—62no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247M16—42no—
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—41no—
266C22⋊C4—244no—
267C22⋊C4—244no—
268C42—41no—
269C42—122no—
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—88no—
277C22×C3—82no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3#39211yescharacteristic
281C22×C3#45512yes—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4#45412yes—
290Dic3—82no—
291Dic3—81no—
292Dic3—81no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—44no—
297C23#93711yescharacteristic
298Q8—42no—
299Q8—81no—
300D4—44no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4—82no—
305D4—82no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4#93711yescharacteristic
319C2×C4#93811yescharacteristic
320C2×C4#93911yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#94011yescharacteristic
323C2×C4#95311yescharacteristic
324C8—42no—
325C8—122no—
326C8—241no—
327C2×C3—22no—
328C2×C3—162no—
329C2×C3#200311yescharacteristic
330C2×C3#200411yescharacteristic
331C2×C3#200811yescharacteristic
332C22—21no—
333C22—21no—
334C22—88no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22#506112yes—
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—242no—
348C4—241no—
349C4—241no—
350C4#520112yes—
351C3#1009711yescharacteristic, complemented, Sylow
352C2—22no—
353C2—162no—
354C2#7173111yescharacteristic
355C2#7209611yescharacteristic
356C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3138C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#3138D3
hypocenterC3#10097
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #83326C2semidirect
4. #71836C2semidirect
6. #69863C2semidirect
7. #67665C2semidirect
351. C3#10097semidirect

Maximal subgroups

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

subgroupindex
C3×#31382
#833262
#831802
#718362
#717602
#698632
#676652
#100973