Finite Group Databasebeta

← #10697332

Subgroups — #10697332

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
1C3×#3165C211yesmaximal, characteristic
2#83373C211yesmaximal, characteristic
3#83224C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic
8#10203—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
17C2×#835—22no—
18#5093—21no—
19#5091—22no—
20#5091—21no—
21#5090—21no—
22#5078—21no—
23#5051—22no—
24#5031—21no—
25#5021—21no—
26#4970—22no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30#4555—22no—
31#4538—22no—
32#3249—31no—
33#3230—31no—
34#3165D311yescharacteristic
35#1986—31no—
36#1981—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⋊Q8D411yescharacteristic
45C3×C4⋊Q8D412yes—
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—22no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59#924—21no—
60#924—21no—
61C2×C4×Dic3—21no—
62C22×C3⋊C8D412yes—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#107—24no—
69C3×#106—24no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#835—22no—
73#835—44no—
74C3⋊#95—21no—
75#830D412yes—
76#810—24no—
77#810—22no—
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⋊Q8C2×D311yescharacteristic
87#501—22no—
88#500—21no—
89#500—61no—
90#498—21no—
91#487—61no—
92#484—62no—
93#477—61no—
94#477—62no—
95#477—61no—
96#467—62no—
97#462—61no—
98#462—61no—
99#431—31no—
100#428—22no—
101C2×C4⋊C8—31no—
102C2×#103—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#339—62no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—41no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×M16—22no—
121C3×M16Q1614yescomplemented
122C2×C3×C8—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140#216—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C4×Dic3—42no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—42no—
154C22×Q8—21no—
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#107—64no—
186#106—24no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—62no—
191#103—64no—
192#103—62no—
193#103—124no—
194C22⋊C8—32no—
195#97—21no—
196C4×C8—21no—
197C4×C8—61no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C3×Q8—44no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—81no—
204C3×Q8—81no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4C2×Q1612yes—
220C2×C3×C4#13712yes—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226Dic6—42no—
227Dic6—41no—
228Dic6—41no—
229C3×C8—24no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—61no—
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—22no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C22⋊C4—121no—
272C42—41no—
273C42—121no—
274C42—122no—
275C42D3×D411yescharacteristic
276C42#23711yescharacteristic
277C42C2×#6711yescharacteristic
278C42C2×#6711yescharacteristic
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—44no—
299Q8—41no—
300Q8—41no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304Q8—122no—
305Q8—121no—
306Q8—121no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—122no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4#93711yescharacteristic
320C2×C4#93711yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#93911yescharacteristic
324C2×C4#94212yes—
325C2×C4C2×#21612yes—
326C2×C4#95311yescharacteristic
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#508611yescharacteristic
338C22#509211yescharacteristic
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—81no—
345C4—81no—
346C4—242no—
347C4—241no—
348C4—241no—
349C4#509012yes—
350C4#519811yescharacteristic
351C4#520111yescharacteristic
352C4#522812yes—
353C3#1020311yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7179311yescharacteristic
357C2#7209811yescharacteristic
358C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3165D3
hypocenterC3#10203
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
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
C3×#31652
#833732
#832242
#718352
#717782
#698682
#676682
#102033