Finite Group Databasebeta

← #10690260

Subgroups — #10690260

Order 768.

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 95 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#79314C211yesmaximal, characteristic
2#73397C211yesmaximal, characteristic
3#70802C211yesmaximal, characteristic
4C4×#824C211yesmaximal, characteristic
5#69857C211yesmaximal, characteristic
6C3×#1300C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic
8#9030—31nomaximal, Sylow
9C2×C4×Dic6C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14C2×#824—21no—
15C4×C4⋊Dic3—21no—
16C4×C4⋊Dic3—21no—
17#4926C2211yescharacteristic
18C4×#208C2211yescharacteristic
19#4828—22no—
20#4719—21no—
21#4718—21no—
22#4701—22no—
23#4661—21no—
24#4661C411yescharacteristic
25#4661C411yescharacteristic
26#4656—21no—
27#4656C411yescharacteristic
28#4656C411yescharacteristic
29#4554—21no—
30#4484—22no—
31#4456—22no—
32#2983—31no—
33#2399—31no—
34C4×#101—31no—
35#1491—31no—
36#1491—31no—
37#1300—31no—
38#1300D311yescharacteristic
39C22×Dic6C2×C411yescharacteristic
40C22×Dic6D411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4C2×C411yescharacteristic
52C2×Dic3⋊C4C2×C411yescharacteristic
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8D412yes—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70C3×#95—21no—
71C3×#95C2×C411yescharacteristic
72C3×#95C2×C411yescharacteristic
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78#824—24no—
79#815—22no—
80Dic3⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83C4⋊(C3⋊C8)D412yes—
84#803—21no—
85#803—21no—
86C4×C3⋊C8—21no—
87C2×C4×Q8—31no—
88C2×C4×Q8—31no—
89#446—62no—
90#446—62no—
91#432—31no—
92#432—61no—
93#432C2×D311yescharacteristic
94#431—61no—
95C2×#101—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#398—31no—
99#398—31no—
100#398—61no—
101#395—31no—
102#395—31no—
103#395—61no—
104C4×C4⋊C4—61no—
105C4×C4⋊C4—61no—
106#376C2×D311yescharacteristic
107#343—61no—
108Q8⋊C8—62no—
109Q8⋊C8—62no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—22no—
126C2×Dic6—42no—
127C2×Dic6—41no—
128C3×M16—22no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C42—24no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—42no—
142C4⋊Dic3—42no—
143C4⋊Dic3—42no—
144C4⋊Dic3—42no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157#208—22no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C22×Q8—31no—
162C22×Q8—31no—
163C2×M16—21no—
164C2×M16—61no—
165C4×Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C42—21no—
183C2×C42—31no—
184C2×C42—31no—
185C2×C42—61no—
186C2×C42C22×D311yescharacteristic
187C4⋊C8—32no—
188C4⋊C8—62no—
189C4⋊C8—62no—
190C4⋊C8D1212yes—
191#101—64no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—21no—
195#97—21no—
196#97—62no—
197#97—61no—
198#97—61no—
199C4×C8—21no—
200C4×C8—62no—
201C4×C8—61no—
202#95—21no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95C4×D311yescharacteristic
208#95C4×D311yescharacteristic
209C23×C3C2×C22⋊C411yescharacteristic
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—22no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3#10412yes—
242Dic6—22no—
243Dic6—22no—
244Dic6—42no—
245Dic6—42no—
246Dic6—41no—
247Dic6—41no—
248C3×C8—42no—
249C3×C8—42no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252C2×Q8—62no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—121no—
258C2×Q8—122no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4C2×C4×D311yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270M16—22no—
271M16—41no—
272M16—62no—
273M16—121no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—62no—
277C2×C8—121no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—122no—
287C4⋊C4—122no—
288C4⋊C4—122no—
289C4⋊C4—122no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C42—24no—
293C42—61no—
294C42—61no—
295C42—61no—
296C42—61no—
297C42—121no—
298C42—121no—
299C42C2×D1211yescharacteristic
300C42#23511yescharacteristic
301C42D3×D411yescharacteristic
302C42#23911yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#38611yescharacteristic
306C22×C3C2×#10411yescharacteristic
307C22×C3#42111yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4#45612yes—
316Dic3—22no—
317Dic3—41no—
318Dic3—42no—
319Dic3—41no—
320Dic3—41no—
321Dic3—81no—
322C23D3×C22⋊C411yescharacteristic
323Q8—62no—
324Q8—62no—
325Q8—122no—
326Q8—122no—
327Q8—121no—
328Q8—121no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—32no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4C4×D1211yescharacteristic
356C2×C4#88211yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#89411yescharacteristic
360C2×C4#89511yescharacteristic
361C8—42no—
362C8—42no—
363C8—122no—
364C8—122no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#187111yescharacteristic
368C2×C3#188611yescharacteristic
369C2×C3#189111yescharacteristic
370C22—21no—
371C22—21no—
372C22#466511yescharacteristic
373C22D3×#10411yescharacteristic
374C22#482111yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—42no—
382C4—62no—
383C4—121no—
384C4—122no—
385C4—121no—
386C4—121no—
387C4—241no—
388C4#482612yes—
389C3#903011yescharacteristic, complemented, Sylow
390C2—21no—
391C2—21no—
392C2#6875511yescharacteristic
393C2#6876511yescharacteristic
394C2#6903611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9030
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
389. C3#9030semidirect

Maximal subgroups

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

subgroupindex
#793142
#733972
#708022
C4×#8242
#698572
C3×#13002
#676152
#90303