Finite Group Databasebeta

← #10698350

Subgroups — #10698350

Order 768.

464 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 89 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#73539C211yesmaximal, characteristic, complemented
2#69944C211yesmaximal, characteristic
3C2×#4470C211yesmaximal, characteristic
4#69456C211yesmaximal, characteristic
5C3×#1282C211yesmaximal, characteristic, complemented
6#67676C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#10376—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×C3×C22⋊C8C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5212C2211yescharacteristic
14#5212C2211yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16C2×#806—21no—
17C2×#806—21no—
18#4946—21no—
19#4942—21no—
20#4933—21no—
21#4933—21no—
22#4660—21no—
23#4657—21no—
24#4470—21no—
25#4470—21no—
26#4470—21no—
27#4470—21no—
28#2569—31no—
29#1964—31no—
30#1519—31no—
31C2×#327—31no—
32#1294—31no—
33#1289—31no—
34#1282D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3—21no—
38C22×D12Q811yescharacteristic
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42—21no—
44C2×#217—21no—
45C2×#217—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3Q811yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#806—42no—
74#806—42no—
75C2×#121—31no—
76C2×#99—61no—
77C2×#99—61no—
78C2×C22⋊C8C2×D311yescharacteristic
79C2×C22⋊C8C2×D311yescharacteristic
80#400—61no—
81#398—61no—
82#387—61no—
83#387—61no—
84#385—61no—
85#380—31no—
86#380—31no—
87#380—61no—
88C4×C22⋊C4C2×D311yescharacteristic
89C2×#95—31no—
90#327—61no—
91#327—61no—
92#327—61no—
93#327—61no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—81no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×Q811yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67—41no—
109C2×#67—41no—
110C2×#67—41no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—42no—
117C22×Dic3—41no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C22×Dic3—81no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—81no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—81no—
127C2×C3×C8—81no—
128C3×C22⋊C4—21no—
129C3×C22⋊C4—21no—
130C3×C22⋊C4—21no—
131C3×C22⋊C4—21no—
132C3×C42—81no—
133#217—41no—
134#217—41no—
135#217—41no—
136#217—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—81no—
142C4⋊Dic3—81no—
143Dic3⋊C4—42no—
144Dic3⋊C4—42no—
145Dic3⋊C4—81no—
146Dic3⋊C4—81no—
147C22×D4—31no—
148C22×D4—61no—
149C23×C4—61no—
150C23×C4C22×D311yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153#121—61no—
154#121—61no—
155#121—61no—
156#121—61no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4D1211yescharacteristic
168C2×C22⋊C4D1211yescharacteristic
169C2×C42—21no—
170#99—122no—
171#99—122no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176#95—21no—
177#95—31no—
178#95—31no—
179#95—31no—
180#95—31no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3#12011yescharacteristic
190C23×C3C22⋊Q811yescharacteristic
191C23×C3#12411yescharacteristic
192C23×C3#13511yescharacteristic
193C23×C3#13511yescharacteristic
194C23×C3AGL(1,ℤ/8)11yescharacteristic
195C23×C3AGL(1,ℤ/8)11yescharacteristic
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—81no—
200C22×D3—81no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4C22⋊Q811yescharacteristic
214C2×C3×C4C22⋊Q811yescharacteristic
215C2×C3×C4#12411yescharacteristic
216C2×C3×C4#12411yescharacteristic
217#67—81no—
218#67—81no—
219#67—81no—
220#67—81no—
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—82no—
237C2×Dic3—82no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241D12—81no—
242D12—81no—
243C3×C8—81no—
244C3×C8—81no—
245C24—61no—
246C24C2×D1211yescharacteristic
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—241no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—81no—
264C22×C4—122no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—241no—
269C22×C4#23511yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4D3×Q811yescharacteristic
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—122no—
282C4⋊C4—122no—
283C4⋊C4—241no—
284C4⋊C4—241no—
285C4⋊C4—241no—
286C4⋊C4—241no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C42—81no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3#39311yescharacteristic
309C22×C3#44911yescharacteristic
310C22×C3#44911yescharacteristic
311C22×C3#47811yescharacteristic
312C22×C3#47811yescharacteristic
313C22×C3#48811yescharacteristic
314C22×C3#49011yescharacteristic
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—161no—
319C2×D3—161no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—81no—
325C3×C4—81no—
326Dic3—42no—
327Dic3—81no—
328Dic3—81no—
329Dic3—81no—
330Dic3—81no—
331Dic3—161no—
332C23—41no—
333C23—41no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—241no—
338C23—241no—
339C23#87511yescharacteristic
340C23#88211yescharacteristic
341C23#89711yescharacteristic
342C23#90411yescharacteristic
343C23#90411yescharacteristic
344C23#90811yescharacteristic
345C23#90811yescharacteristic
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350D4—241no—
351D4—241no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—62no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—242no—
380C2×C4—242no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4#88511yescharacteristic
385C2×C4#88511yescharacteristic
386C2×C4#89611yescharacteristic
387C2×C4#89611yescharacteristic
388C8—81no—
389C8—81no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3#202511yescharacteristic
397C2×C3#203111yescharacteristic
398C2×C3#203211yescharacteristic
399D3—161no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—241no—
411C22—241no—
412C22—241no—
413C22—481no—
414C22—481no—
415C22#467311yescharacteristic
416C22#469711yescharacteristic
417C22#471311yescharacteristic
418C22#473411yescharacteristic
419C22#473411yescharacteristic
420C22#484711yescharacteristic
421C22#484711yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—81no—
428C4—122no—
429C4—241no—
430C4—241no—
431C4—241no—
432C4—241no—
433C4—481no—
434C3#1037611yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—481no—
442C2#6867911yescharacteristic
443C2#6896711yescharacteristic
444C2#6904211yescharacteristic
subgroupG/H
centerC22#4673
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#393
prime radicalC2×#930C22
hypercenter#1282D3
hypocenterC3#10376
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
1. #73539C2semidirect
5. C3×#1282C2semidirect
6. #67676C2semidirect
7. #67675C2semidirect
34. #1282D3semidirect
434. C3#10376semidirect

Maximal subgroups

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

subgroupindex
#735392
#699442
C2×#44702
#694562
C3×#12822
#676762
#676752
#103763