Finite Group Databasebeta

← #10697654

Subgroups — #10697654

Order 768.

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

normal subgroups only

This group has 504 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#83243C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#79068C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic
6#70767C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10253—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5230—22no—
15#5121—21no—
16#5120—21no—
17#5093—21no—
18#5091—22no—
19#5078—21no—
20#4975—21no—
21#4973—21no—
22#4972—22no—
23#4970—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4851—21no—
27#4850—21no—
28#4829—22no—
29C3×#332—22no—
30#4467—22no—
31#4456—22no—
32#3230—31no—
33#3078—31no—
34#2567—31no—
35#1549D311yescharacteristic
36#1503—31no—
37#1479—31no—
38#1303—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⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52#924—21no—
53#924—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—24no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58#870—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3⋊#95—24no—
73C3⋊#95D412yes—
74#830—21no—
75#818—22no—
76#817—22no—
77#816—22no—
78#816—22no—
79#810—24no—
80#807—24no—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×Q8—31no—
86#500—61no—
87#487—61no—
88#484—62no—
89#479—62no—
90#477—61no—
91#477—62no—
92#475—61no—
93#431—31no—
94#431C2×D311yescharacteristic
95#428—61no—
96#423C2×D311yescharacteristic
97#419—61no—
98#395—62no—
99#391—61no—
100#389—62no—
101#384—61no—
102C4×C4⋊C4—31no—
103#332—22no—
104#328—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6Q1612yes—
124C3×M16—41no—
125C3×M16—41no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137C4⋊Dic3—24no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—84no—
149C4×Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C22×Q8—21no—
156C22×Q8—31no—
157C2×M16—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—64no—
179C2×C4⋊C4D1212yes—
180C2×C42—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—64no—
184#107—62no—
185#106—62no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8#6712yes—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C4×C8—21no—
196C4×C8—61no—
197#95—32no—
198#95—64no—
199C23×C3#12011yescharacteristic
200C3×Q8—82no—
201C3×Q8—82no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—44no—
205C2×C3×C4—42no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12711yescharacteristic
212C2×C3×C4C2×Q1611yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—24no—
217C2×Dic3—22no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226Dic6—22no—
227Dic6—22no—
228Dic6—41no—
229Dic6—41no—
230Dic6—42no—
231Dic6—81no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C2×Q8—32no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—62no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C2×D1211yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253M16—41no—
254M16—41no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—62no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—64no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—244no—
274C22⋊C4—121no—
275C42—121no—
276C42—121no—
277C42—121no—
278C42—121no—
279C42—121no—
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#39211yescharacteristic
287C22×C3#45111yescharacteristic
288C22×C3#45211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—82no—
293C3×C4#45111yescharacteristic
294C3×C4#45211yescharacteristic
295C3×C4#45612yes—
296C3×C4#46711yescharacteristic
297C3×C4#47011yescharacteristic
298Dic3—44no—
299Dic3—42no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23#88211yescharacteristic
304Q8—62no—
305Q8—62no—
306Q8—82no—
307Q8—82no—
308Q8—121no—
309Q8—121no—
310Q8—122no—
311Q8—241no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—64no—
317C2×C4—62no—
318C2×C4—122no—
319C2×C4—121no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4#88211yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4#88411yescharacteristic
329C2×C4#91511yescharacteristic
330C2×C4#91611yescharacteristic
331C2×C4D3×Q1611yescharacteristic
332C2×C4#91811yescharacteristic
333C2×C4#93011yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#94011yescharacteristic
336C8—42no—
337C8—81no—
338C8—122no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#200111yescharacteristic
342C2×C3#200511yescharacteristic
343C2×C3#200811yescharacteristic
344C22—21no—
345C22—21no—
346C22#480811yescharacteristic
347C22#480911yescharacteristic
348C22#496011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—124no—
354C4—122no—
355C4—241no—
356C4—241no—
357C4—241no—
358C4#475212yes—
359C4#481311yescharacteristic
360C4#481611yescharacteristic
361C4#517311yescharacteristic
362C4#518611yescharacteristic
363C3#1025311yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2#7011911yescharacteristic
367C2#7027311yescharacteristic
368C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1549D3
hypocenterC3#10253
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
363. C3#10253semidirect

Maximal subgroups

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

subgroupindex
#832432
#832242
#790682
C3×#15492
#707922
#707672
#677152
#102533