Finite Group Databasebeta

← #10697221

Subgroups — #10697221

Order 768.

496 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 101 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#83223C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic
3#79068C211yesmaximal, characteristic
4C3×#1546C211yesmaximal, characteristic
5#70786C211yesmaximal, characteristic
6#70768C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic
8#10187—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11#5470C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#835—21no—
16#5121—21no—
17#5120—21no—
18#5094—22no—
19#5093—21no—
20#5090—22no—
21#5090—22no—
22#5080—21no—
23#4971—22no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4851—21no—
27#4850—21no—
28#4829—22no—
29C3×#332—22no—
30#4469—22no—
31#4456—22no—
32#3229—31no—
33#3078—31no—
34C2×#462—31no—
35#1546D311yescharacteristic
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×Dic6D411yescharacteristic
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—24no—
50C2×#216—42no—
51C2×#216—42no—
52#926—21no—
53C2×C4⋊Dic3D411yescharacteristic
54#924—21no—
55#924—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C22×C3⋊C8—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#835—24no—
74#830D412yes—
75#818—22no—
76#817—22no—
77#816—22no—
78#816—22no—
79#810—24no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C22×Q16—61no—
86#533—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#498—61no—
90#487—61no—
91#483—62no—
92#479—62no—
93#477—61no—
94#475—61no—
95#475—62no—
96#462—62no—
97#462—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#419—61no—
104C2×#103—61no—
105#332—22no—
106Q8⋊C8—62no—
107#324—62no—
108C2×C3×Q8—22no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6SD1612yes—
128C3×M16—41no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#217—41no—
141#216—48no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—21no—
160C22×Q8—31no—
161C2×Q16—64no—
162C2×Q16—122no—
163C2×Q16—122no—
164C2×M16—21no—
165C22×C8—61no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8D1212yes—
172#125—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181#117—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#106—64no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—64no—
198#103—62no—
199#103—62no—
200#103—64no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C4×C8—21no—
204C4×C8—61no—
205C23×C3#12111yescharacteristic
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—81no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—22no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232Dic6—22no—
233Dic6—22no—
234Dic6—41no—
235Dic6—41no—
236Dic6—42no—
237Dic6—42no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C2×Q8—22no—
243C2×Q8—32no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C22×C4—41no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4#23511yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262Q16—128no—
263M16—41no—
264M16—41no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—121no—
270C2×C8—121no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C42—121no—
287C42—121no—
288C42—121no—
289C42C2×D1211yescharacteristic
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42C2×#6711yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#39211yescharacteristic
296C22×C3#46511yescharacteristic
297C22×C3#46811yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4#44811yescharacteristic
303C3×C4#44911yescharacteristic
304C3×C4#45011yescharacteristic
305C3×C4#45211yescharacteristic
306C3×C4#47312yes—
307Dic3—42no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23#88411yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—62no—
315Q8—62no—
316Q8—81no—
317Q8—81no—
318Q8—122no—
319Q8—121no—
320Q8—121no—
321Q8—122no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—62no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—241no—
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#91111yescharacteristic
338C2×C4#91211yescharacteristic
339C2×C4D3×SD1611yescharacteristic
340C2×C4#91511yescharacteristic
341C2×C4#93011yescharacteristic
342C2×C4#93711yescharacteristic
343C2×C4#94011yescharacteristic
344C8—42no—
345C8—81no—
346C8—122no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#200011yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#202011yescharacteristic
353C22—21no—
354C22—21no—
355C22#477811yescharacteristic
356C22#478011yescharacteristic
357C22#496011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—122no—
363C4—241no—
364C4—241no—
365C4—241no—
366C4#475512yes—
367C4#477611yescharacteristic
368C4#478311yescharacteristic
369C4#515811yescharacteristic
370C4#517011yescharacteristic
371C3#1018711yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2#7012511yescharacteristic
375C2#7020511yescharacteristic
376C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1546D3
hypocenterC3#10187
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
371. C3#10187semidirect

Maximal subgroups

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

subgroupindex
#832232
C2×#50902
#790682
C3×#15462
#707862
#707682
#677142
#101873