Finite Group Databasebeta

← #10697337

Subgroups — #10697337

Order 768.

490 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 109 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#79272C211yesmaximal, characteristic
2#79037C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4#70792C211yesmaximal, characteristic
5#70456C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7C3×#1283C211yesmaximal, characteristic
8#10203—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5121—22no—
16#5116—22no—
17C2×#816—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4850—22no—
20#4850—21no—
21#4850—21no—
22#4849—21no—
23#4849—21no—
24#4722—22no—
25#4702—22no—
26#4682—21no—
27#4682—21no—
28#4681—21no—
29#4680—21no—
30#4555—22no—
31#4467—22no—
32#3249—31no—
33#3230—31no—
34#3165—31no—
35#1986—31no—
36#1981—31no—
37#1503—31no—
38#1283D311yescharacteristic
39C22×Dic6—21no—
40C2×C3×M16D412yes—
41C22×C3×C8D412yes—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C2×Dic12—22no—
54#870—21no—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3⋊#95—21no—
76#818—24no—
77#818—24no—
78#818—22no—
79#817—24no—
80#816—24no—
81#816—22no—
82#816—22no—
83#816—44no—
84#533—31no—
85#533—31no—
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#500—61no—
90#498—61no—
91#487—61no—
92#484—62no—
93#477—62no—
94#477—61no—
95#477—61no—
96#467—62no—
97#462—61no—
98#462—61no—
99#431C2×D311yescharacteristic
100#428—62no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#103—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#339—62no—
106#328—62no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—41no—
111C22×Dic3—41no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—22no—
116C2×Dic6—21no—
117C2×Dic6—42no—
118C2×Dic6—42no—
119C2×Dic6—41no—
120C3×M16—22no—
121C3×M16Q1614yescomplemented
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—42no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#217—41no—
131#217—41no—
132C4⋊Dic3—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—42no—
153Dic12—82no—
154C22×Q8—61no—
155C2×Q16—62no—
156C2×M16D1212yes—
157C22×C8D1212yes—
158C4⋊Q8—32no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166#125—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—61no—
177#117—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—64no—
184#107—62no—
185#107—64no—
186#106—64no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190#103—62no—
191#103—62no—
192#103—64no—
193#103—124no—
194C22⋊C8D1212yes—
195#97—21no—
196C4×C8—21no—
197C4×C8—21no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4C2×Q1612yes—
210C2×C3×C4#13712yes—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—82no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220Dic6—42no—
221Dic6—44no—
222Dic6—41no—
223Dic6—41no—
224Dic6—41no—
225Dic6—41no—
226Dic6—81no—
227Dic6—81no—
228Dic6—81no—
229C3×C8—24no—
230C3×C8—42no—
231C3×C8—42no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—62no—
236C2×Q8—61no—
237C2×Q8—122no—
238C2×Q8—122no—
239C2×Q8—121no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4C2×D1211yescharacteristic
243C22×C4C2×D1211yescharacteristic
244C22×C4C2×D1211yescharacteristic
245Q16—242no—
246M16—22no—
247M16Dic1214yescomplemented
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—42no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—121no—
273C42—121no—
274C42—122no—
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42#23911yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#39211yescharacteristic
282C22×C3#49411yescharacteristic
283C22×C3#49711yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4#45112yes—
288C3×C4#45311yescharacteristic
289C3×C4#45411yescharacteristic
290C3×C4#46212yes—
291Dic3—82no—
292Dic3—82no—
293Dic3—81no—
294Dic3—81no—
295Dic3—81no—
296Dic3—81no—
297C23#87411yescharacteristic
298Q8—124no—
299Q8—122no—
300Q8—121no—
301Q8—121no—
302Q8—121no—
303Q8—121no—
304Q8—241no—
305Q8—241no—
306Q8—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—122no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4#88211yescharacteristic
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#89511yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4C2×Dic1212yes—
326C2×C4#90912yes—
327C8—24no—
328C8—42no—
329C8—42no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200611yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201611yescharacteristic
335C22—21no—
336C22—21no—
337C22#467111yescharacteristic
338C22#469911yescharacteristic
339C22#471511yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—242no—
344C4—242no—
345C4—241no—
346C4—241no—
347C4—241no—
348C4—241no—
349C4#473912yes—
350C4#474711yescharacteristic
351C4#475311yescharacteristic
352C4#484912yes—
353C3#1020311yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#6868811yescharacteristic
357C2#6900811yescharacteristic
358C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1283D3
hypocenterC3#10203
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
121. C3×M16Q16semidirect
247. M16Dic12semidirect
353. C3#10203semidirect

Maximal subgroups

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

subgroupindex
#792722
#790372
#785062
#707922
#704562
#698682
C3×#12832
#102033