Finite Group Databasebeta

← #10697335

Subgroups — #10697335

Order 768.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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#83373C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic
5#69486C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic
8#10203—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5121—22no—
15#5091—21no—
16#5090—21no—
17#5079—21no—
18#5078—21no—
19#5051—22no—
20#5044—22no—
21#5031—21no—
22#5021—22no—
23#5021—21no—
24C2×#810—22no—
25#4925C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4702—22no—
28#4682—21no—
29#4680—21no—
30C3×#339—22no—
31#4452—22no—
32#3249—31no—
33#3230—31no—
34#3165—31no—
35#1986D311yescharacteristic
36#1981—31no—
37#1503—31no—
38#1283—31no—
39C22×Dic6—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53#926—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56#924—21no—
57#924—21no—
58C22×C3⋊C8D412yes—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70C3×#95—21no—
71#835—24no—
72#835—22no—
73#830D412yes—
74#818—24no—
75#817—24no—
76#810—22no—
77#810—44no—
78#807—24no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#533C2×D311yescharacteristic
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#431—31no—
100#428—62no—
101C2×C4⋊C8—31no—
102C2×#103—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4C2×D311yescharacteristic
105#339—22no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—41no—
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—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42—42no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139#216—82no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—42no—
154C22×Q8—61no—
155C2×Q16—62no—
156C2×M16#6712yes—
157C22×C8—32no—
158C4⋊Q8—32no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8D1211yescharacteristic
165C4⋊Q8D1211yescharacteristic
166#125—21no—
167#125—61no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—21no—
177#117—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C42—21no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#107—64no—
185#107—64no—
186#106—64no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—62no—
191#103—62no—
192#103—64no—
193#103—124no—
194C22⋊C8—32no—
195#97—21no—
196C4×C8—21no—
197C4×C8—61no—
198#95—21no—
199C23×C3#12711yescharacteristic
200C3×Q8—42no—
201C3×Q8—41no—
202C3×Q8—41no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4C2×Q1612yes—
218C2×C3×C4#13712yes—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—82no—
223Dic6—44no—
224Dic6—41no—
225Dic6—41no—
226Dic6—81no—
227Dic6—81no—
228Dic6—81no—
229C3×C8—24no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—61no—
235C2×Q8—62no—
236C2×Q8—61no—
237C2×Q8—122no—
238C2×Q8—122no—
239C2×Q8—121no—
240C22×C4—41no—
241C22×C4—121no—
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4C2×#6711yescharacteristic
245Q16—242no—
246M16—22no—
247M16#21614yescomplemented
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—122no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C22⋊C4—121no—
272C42—41no—
273C42—42no—
274C42—121no—
275C42C2×D1211yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
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—82no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4#45112yes—
291C3×C4#45311yescharacteristic
292C3×C4#45411yescharacteristic
293C3×C4#46212yes—
294Dic3—82no—
295Dic3—81no—
296Dic3—81no—
297C23#94011yescharacteristic
298Q8—42no—
299Q8—41no—
300Q8—41no—
301Q8—124no—
302Q8—121no—
303Q8—121no—
304Q8—241no—
305Q8—241no—
306Q8—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—242no—
319C2×C4#88211yescharacteristic
320C2×C4#88211yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#93011yescharacteristic
324C2×C4#93711yescharacteristic
325C2×C4C2×#21612yes—
326C2×C4#95112yes—
327C8—24no—
328C8—122no—
329C8—122no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200611yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201611yescharacteristic
335C22—21no—
336C22—21no—
337C22#496011yescharacteristic
338C22#506011yescharacteristic
339C22#509211yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—81no—
345C4—81no—
346C4—241no—
347C4—242no—
348C4—241no—
349C4#482211yescharacteristic
350C4#482711yescharacteristic
351C4#503112yes—
352C4#505012yes—
353C3#1020311yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#6969111yescharacteristic
357C2#7039211yescharacteristic
358C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1986D3
hypocenterC3#10203
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
121. C3×M16Q16semidirect
247. M16#216semidirect
353. C3#10203semidirect

Maximal subgroups

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

subgroupindex
#833732
#833242
#785062
C3×#19862
#694862
#694712
#676682
#102033