Finite Group Databasebeta

← #10697307

Subgroups — #10697307

Order 768.

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

normal subgroups only

This group has 486 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#83370C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic
5#69489C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic
8#10199—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5121—22no—
15#5093—21no—
16#5092—21no—
17#5091—21no—
18#5090—21no—
19#5051—22no—
20#5050—22no—
21#5031—22no—
22#5031—21no—
23#5021—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4702—22no—
27#4682—21no—
28#4680—21no—
29C3×#339—22no—
30#4529—22no—
31#4452—22no—
32#3249—31no—
33#3209—31no—
34#3165—31no—
35#1987—31no—
36#1986D311yescharacteristic
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—24no—
53C2×#216—22no—
54#926—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57#924—21no—
58#924—21no—
59C22×C3⋊C8D412yes—
60#870—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D412yes—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#97—21no—
70C3×C4×C8—21no—
71C3×#95—21no—
72#835—22no—
73#830D412yes—
74#818—24no—
75#817—24no—
76#810—24no—
77#810—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#501—62no—
87#500—61no—
88#498—61no—
89#494—61no—
90#487—61no—
91#477—61no—
92#477—61no—
93#467—62no—
94#462—62no—
95#462—61no—
96#462—61no—
97#451—62no—
98#431—31no—
99#428—62no—
100C2×C4⋊C8—31no—
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4C2×D311yescharacteristic
103#340—62no—
104#339—22no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—41no—
113C2×Dic6—21no—
114C2×Dic6—22no—
115C2×Dic6—21no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C3×M16—22no—
119C3×M16Q1614yescomplemented
120C2×C3×C8—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—41no—
131C3×C42—42no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137#216—84no—
138#216—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C22×Q8—61no—
154C2×Q16—64no—
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#106—64no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—64no—
190#103—62no—
191#103—62no—
192C22⋊C8—32no—
193#97—21no—
194C4×C8—21no—
195C4×C8—61no—
196#95—21no—
197C23×C3#12711yescharacteristic
198C3×Q8—42no—
199C3×Q8—41no—
200C3×Q8—41no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4C2×Q1612yes—
216C2×C3×C4#13712yes—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—82no—
221Dic6—44no—
222Dic6—41no—
223Dic6—41no—
224Dic6—82no—
225Dic6—81no—
226C3×C8—24no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—61no—
234C2×Q8—121no—
235C2×Q8—242no—
236C22×C4—41no—
237C22×C4—121no—
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4C2×#6711yescharacteristic
241Q16—244no—
242Q16—242no—
243M16—22no—
244M16#21614yescomplemented
245C2×C8—22no—
246C2×C8—62no—
247C2×C8—62no—
248C2×C8—122no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—62no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—41no—
268C22⋊C4—121no—
269C42—41no—
270C42—42no—
271C42—121no—
272C42C2×D1211yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#49411yescharacteristic
280C22×C3#49711yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—82no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4#45112yes—
288C3×C4#45511yescharacteristic
289C3×C4#45611yescharacteristic
290C3×C4#46212yes—
291Dic3—81no—
292Dic3—82no—
293Dic3—81no—
294C23#94011yescharacteristic
295Q8—42no—
296Q8—41no—
297Q8—41no—
298Q8—124no—
299Q8—121no—
300Q8—121no—
301Q8—241no—
302Q8—242no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—242no—
315C2×C4#88211yescharacteristic
316C2×C4#88211yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#93011yescharacteristic
320C2×C4#93711yescharacteristic
321C2×C4C2×#21612yes—
322C2×C4#95112yes—
323C8—24no—
324C8—122no—
325C8—122no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#200711yescharacteristic
329C2×C3#200811yescharacteristic
330C2×C3#201611yescharacteristic
331C22—21no—
332C22—21no—
333C22#496011yescharacteristic
334C22#506011yescharacteristic
335C22#509211yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—81no—
341C4—81no—
342C4—242no—
343C4—241no—
344C4—241no—
345C4#482511yescharacteristic
346C4#482611yescharacteristic
347C4#503112yes—
348C4#505012yes—
349C3#1019911yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2#6969111yescharacteristic
353C2#7039311yescharacteristic
354C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1986D3
hypocenterC3#10199
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
119. C3×M16Q16semidirect
244. M16#216semidirect
349. C3#10199semidirect

Maximal subgroups

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

subgroupindex
#833732
#833702
#785062
C3×#19862
#694892
#694712
#676682
#101993