Finite Group Databasebeta

← #10713839

Subgroups — #10713839

Order 768.

484 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 91 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#74769C211yesmaximal, characteristic
2Dic3×#100C211yesmaximal, characteristic
3#70028C211yesmaximal, characteristic
4C3×#1390C211yesmaximal, characteristic
5#67698C211yesmaximal, characteristic
6#67694C211yesmaximal, characteristic
7#67637C211yesmaximal, characteristic
8#13294—31nomaximal, Sylow
9C2×D4×Dic3C2211yescharacteristic
10C22×C4⋊Dic3C411yescharacteristic, complemented
11C2×C3×#100C2211yescharacteristic
12C2×C3×#99C2211yescharacteristic
13C3×#386C2211yescharacteristic
14#5220C2211yescharacteristic
15C2×#834C2211yescharacteristic
16C2×#833C2211yescharacteristic
17#5210C411yescharacteristic, complemented
18Dic3×M16—21no—
19#4950—21no—
20#4950C411yescharacteristic, complemented
21#4948—21no—
22#4942—21no—
23#4942C411yescharacteristic, complemented
24#4661—21no—
25#4655—21no—
26#4554—21no—
27#4527—21no—
28#2579—31no—
29#1799—31no—
30C4×#100—31no—
31#1390—31no—
32#1390—31no—
33#1390—31no—
34#1390D311yescharacteristic
35C22×C3×D4C2311yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×Dic3C2×C411yescharacteristic
38C23×Dic3C2×C411yescharacteristic
39C23×Dic3D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217C2×C411yescharacteristic
47C2×#217C2×C411yescharacteristic
48C2×#217D411yescharacteristic
49D4×Dic3—41no—
50D4×Dic3—41no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—41no—
55C2×C4⋊Dic3—41no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—41no—
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62C3×#100—21no—
63C3×#100Q812yes—
64C3×#99—21no—
65C3×#99—21no—
66C3×#95—21no—
67#834—21no—
68#834—21no—
69#833—21no—
70#833Q812yes—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—41no—
76#815—41no—
77C8×Dic3—41no—
78C2×C4×D4—31no—
79C22×C4⋊C4—31no—
80C2×#100—31no—
81C2×#100C2×D311yescharacteristic
82C2×#99—31no—
83C2×#99C2×D311yescharacteristic
84C4×M16—61no—
85#398—61no—
86#396—61no—
87#387—61no—
88#386—31no—
89#386C2×D311yescharacteristic
90#385—31no—
91#385—31no—
92#385—61no—
93#380—31no—
94#380—61no—
95#343—61no—
96#342—61no—
97C24×C3C22×C411yescharacteristic
98C24×C3C2×D411yescharacteristic
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4C2×Q811yescharacteristic
102C2×C3×D4C2×Q811yescharacteristic
103C2×C3×D4#4111yescharacteristic
104C2×C3×D4#4111yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—42no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—41no—
118C22×Dic3—41no—
119C22×Dic3—81no—
120C22×Dic3—81no—
121C22×Dic3—81no—
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C3×M16—22no—
127C3×M16—41no—
128C2×C3×C8—41no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133#217—41no—
134#217—41no—
135#217—41no—
136#217—41no—
137#217—41no—
138#217—81no—
139C4⋊Dic3—41no—
140C4⋊Dic3—42no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—81no—
144Dic3⋊C4—42no—
145Dic3⋊C4—81no—
146Dic3⋊C4—82no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149#208—22no—
150#208—41no—
151C2×C3⋊C8—41no—
152C22×D4C22×D311yescharacteristic
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4C4×D311yescharacteristic
157C2×M16—21no—
158C2×M16—61no—
159C4×D4—121no—
160C4×D4—121no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—121no—
168C2×C4⋊C4—121no—
169C2×C4⋊C4—121no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4C4×D311yescharacteristic
178C2×C42—31no—
179#100—21no—
180#100—32no—
181#100—61no—
182#100Dic612yes—
183#99—21no—
184#99—21no—
185#99—61no—
186#99—61no—
187#97—121no—
188C4×C8—121no—
189#95—21no—
190#95—61no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95—121no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3C4×Q811yescharacteristic
205C23×C3C4×Q811yescharacteristic
206C23×C3#12411yescharacteristic
207C23×C3C4⋊Q811yescharacteristic
208C3×D4—41no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4C22⋊Q811yescharacteristic
226C2×C3×C4C22⋊Q811yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—82no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C2×Dic3#9911yescharacteristic
249C2×Dic3#9911yescharacteristic
250C3×C8—42no—
251C3⋊C8—42no—
252C24C2×C4×D311yescharacteristic
253C24D3×D411yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4C2×Dic611yescharacteristic
257C2×D4#23511yescharacteristic
258C2×D4#23711yescharacteristic
259C2×D4D3×Q811yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—241no—
278C22×C4—241no—
279C22×C4—241no—
280C22×C4D3×D411yescharacteristic
281M16—22no—
282M16—41no—
283M16—62no—
284M16—121no—
285C2×C8—41no—
286C2×C8—121no—
287C4⋊C4—121no—
288C4⋊C4—122no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—122no—
292C4⋊C4—241no—
293C4⋊C4—241no—
294C4⋊C4—242no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—241no—
305C42—61no—
306C42—61no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—42no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—81no—
323C22×C3#39111yescharacteristic
324C22×C3C2×#9911yescharacteristic
325C22×C3#41011yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—81no—
330C3×C4—161no—
331Dic3—21no—
332Dic3—21no—
333Dic3—41no—
334Dic3—41no—
335Dic3—41no—
336Dic3—81no—
337Dic3—81no—
338Dic3—161no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C4×Dic611yescharacteristic
348C23D3×C22⋊C411yescharacteristic
349C23#88511yescharacteristic
350C23#88711yescharacteristic
351C23#88811yescharacteristic
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—242no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4#87811yescharacteristic
393C2×C4#89611yescharacteristic
394C8—42no—
395C8—122no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3#199611yescharacteristic
403C2×C3#212011yescharacteristic
404C2×C3#212111yescharacteristic
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22#464611yescharacteristic
422C22#474011yescharacteristic
423C22D3×#9911yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—61no—
428C4—61no—
429C4—81no—
430C4—121no—
431C4—121no—
432C4—121no—
433C4—161no—
434C4—241no—
435C4—241no—
436C4—481no—
437C3#1329411yescharacteristic, complemented, Sylow
438C2—21no—
439C2—21no—
440C2—41no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2#6895511yescharacteristic
445C2#6913611yescharacteristic
446C2#6914311yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1390C2
socleC22×C3C2×#99
prime radical, cosocleC22×C3×D4C23
hypercenter#1390D3
hypocenterC3#13294
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
10. C22×C4⋊Dic3C4semidirect
17. #5210C4semidirect
20. #4950C4semidirect
23. #4942C4semidirect
437. C3#13294semidirect

Maximal subgroups

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

subgroupindex
#747692
Dic3×#1002
#700282
C3×#13902
#676982
#676942
#676372
#132943