Finite Group Databasebeta

← #10697304

Subgroups — #10697304

Order 768.

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

normal subgroups only

This group has 486 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
1C3×#3165C211yesmaximal, characteristic
2#83373C211yesmaximal, characteristic
3#83222C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#71779C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic
8#10199—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5228—22no—
18#5092—21no—
19#5091—21no—
20#5090—22no—
21#5090—21no—
22#5079—21no—
23#5051—22no—
24#5031—21no—
25#5021—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4555—22no—
30#4538—22no—
31#4529—22no—
32#3249—31no—
33#3209—31no—
34#3165D311yescharacteristic
35#1987—31no—
36#1986—31no—
37#1503—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#216—24no—
57C2×#216—22no—
58#926—21no—
59C2×C4⋊Dic3—21no—
60#924—21no—
61#924—21no—
62C2×C4×Dic3—21no—
63C22×C3⋊C8D412yes—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#107—24no—
70C3×#106—24no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#835—24no—
74#835—22no—
75C3⋊#95—21no—
76#830D412yes—
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⋊Q8C2×D311yescharacteristic
86#501—22no—
87#500—21no—
88#498—21no—
89#494—61no—
90#487—61no—
91#477—61no—
92#477—61no—
93#467—62no—
94#462—61no—
95#462—62no—
96#462—61no—
97#451—62no—
98#431—31no—
99#428—22no—
100C2×C4⋊C8—31no—
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—41no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C3×M16—22no—
119C3×M16Q1614yescomplemented
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138#216—84no—
139#216—82no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149C4×Dic3—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C22×Q8—21no—
154C2×Q16—64no—
155C2×Q16—62no—
156C2×M16#6712yes—
157C22×C8—32no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8#6712yes—
164C4⋊Q8#6711yescharacteristic
165C4⋊Q8#6711yescharacteristic
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—21no—
180C2×C4⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—24no—
184#107—62no—
185#106—24no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—62no—
190#103—64no—
191#103—62no—
192C22⋊C8—32no—
193#97—21no—
194C4×C8—21no—
195C4×C8—61no—
196#95—61no—
197C23×C3#12711yescharacteristic
198C3×Q8—44no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—81no—
202C3×Q8—82no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—82no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4C2×Q1612yes—
217C2×C3×C4#13712yes—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223Dic6—42no—
224Dic6—41no—
225Dic6—41no—
226C3×C8—24no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229C2×Q8—21no—
230C2×Q8—22no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—82no—
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—21no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—41no—
268C22⋊C4—121no—
269C42—41no—
270C42—121no—
271C42—122no—
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
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—81no—
285C3×C4—82no—
286C3×C4—81no—
287C3×C4#45112yes—
288C3×C4#45511yescharacteristic
289C3×C4#45611yescharacteristic
290C3×C4#46212yes—
291Dic3—82no—
292Dic3—81no—
293Dic3—81no—
294C23#94011yescharacteristic
295Q8—44no—
296Q8—41no—
297Q8—41no—
298Q8—81no—
299Q8—82no—
300Q8—122no—
301Q8—121no—
302Q8—121no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4#93711yescharacteristic
316C2×C4#93711yescharacteristic
317C2×C4#93811yescharacteristic
318C2×C4#93911yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94212yes—
321C2×C4C2×#21612yes—
322C2×C4#95311yescharacteristic
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#508611yescharacteristic
334C22#509211yescharacteristic
335C22#522311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—81no—
341C4—81no—
342C4—242no—
343C4—241no—
344C4—241no—
345C4#509012yes—
346C4#520211yescharacteristic
347C4#520511yescharacteristic
348C4#522812yes—
349C3#1019911yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2#7179311yescharacteristic
353C2#7209911yescharacteristic
354C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3165D3
hypocenterC3#10199
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

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
C3×#31652
#833732
#832222
#718352
#717792
#698682
#676682
#101993