Finite Group Databasebeta

← #10698671

Subgroups — #10698671

Order 768.

460 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C3×#2820C211yesmaximal, characteristic
2#71741C211yesmaximal, characteristic, complemented
3C2×#4543C211yesmaximal, characteristic, complemented
4#71726C211yesmaximal, characteristic
5C2×#4540C211yesmaximal, characteristic
6#69550C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#10427—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#399—21no—
12C3×#397—21no—
13C3×#396—21no—
14C3×#396—21no—
15C3×#395—21no—
16C3×#394—21no—
17C3×#382C2211yescharacteristic
18C2×#835—21no—
19C2×#832—21no—
20C2×#808—21no—
21C2×#807—21no—
22#4935C2211yescharacteristic
23C2×#803C2211yescharacteristic
24C2×#803C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4543—22no—
27#4540—22no—
28#4529—21no—
29#4529—21no—
30#2820D311yescharacteristic
31#1992—31no—
32#1959—31no—
33#1847—31no—
34C2×#330—31no—
35C2×#329—31no—
36#1295—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C3×#125—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72#835—42no—
73#832—42no—
74#808—42no—
75#807—42no—
76#803—21no—
77#803—21no—
78#803—21no—
79#803—21no—
80#803—21no—
81#803D412yes—
82C2×#125C2×D311yescharacteristic
83C2×#124C2×D311yescharacteristic
84C2×#107—61no—
85C2×#106—61no—
86C2×#103—61no—
87C2×#102—61no—
88C2×#97—31no—
89C2×#97—31no—
90C2×#97—31no—
91#399—21no—
92#397—21no—
93#396—21no—
94#396—21no—
95#395—21no—
96#394—21no—
97#384—31no—
98#382C2×D311yescharacteristic
99#340—61no—
100#340—61no—
101#330—62no—
102#329—62no—
103C24×C3—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—81no—
132C3×C4⋊C4—81no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—81no—
138C3×C22⋊C4—81no—
139C3×C22⋊C4—81no—
140C3×C22⋊C4—81no—
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8#6711yescharacteristic
160C22×D4#6711yescharacteristic
161C22×C8—61no—
162C22×C8—61no—
163C22×C8—61no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#124—21no—
169#124—21no—
170#124—21no—
171#124—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4#6711yescharacteristic
180C2×C4⋊C4#6711yescharacteristic
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6711yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C42C22×D311yescharacteristic
190#107—122no—
191#106—122no—
192#103—122no—
193#102—122no—
194#97—32no—
195#97—61no—
196#97—61no—
197#97—61no—
198#97—61no—
199#97—61no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3#12011yescharacteristic
212C3×Q8—42no—
213C3×Q8—81no—
214C3×D4—42no—
215C3×D4—81no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×C3×C4#12411yescharacteristic
243C2×Dic3—81no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C3⋊C8—24no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—121no—
264C22×C4C2×#6711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266C22×C4C2×#6711yescharacteristic
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8—121no—
274C2×C8—121no—
275C2×C8—121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—122no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C42D3×D411yescharacteristic
302C42#23711yescharacteristic
303C42#23711yescharacteristic
304C42#23711yescharacteristic
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3#39411yescharacteristic
313C22×C3#45311yescharacteristic
314C22×C3#45311yescharacteristic
315C22×C3#45411yescharacteristic
316C22×C3#45511yescharacteristic
317C22×C3#45611yescharacteristic
318C22×C3#45611yescharacteristic
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4—81no—
330Dic3—161no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23#95311yescharacteristic
339Q8—42no—
340Q8—81no—
341D4—42no—
342D4—81no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
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—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#93511yescharacteristic
368C2×C4#93511yescharacteristic
369C2×C4#93611yescharacteristic
370C2×C4#93811yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#93911yescharacteristic
373C8—64no—
374C8—122no—
375C8—122no—
376C2×C3—161no—
377C2×C3#165611yescharacteristic
378C2×C3#166011yescharacteristic
379C2×C3#166111yescharacteristic
380C2×C3#166411yescharacteristic
381C2×C3#203711yescharacteristic
382C2×C3#203811yescharacteristic
383C2×C3#203911yescharacteristic
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#522211yescharacteristic
392C22#524111yescharacteristic
393C22#524111yescharacteristic
394C22#524211yescharacteristic
395C22#524511yescharacteristic
396C22#524611yescharacteristic
397C22#524611yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—481no—
410C3#1042711yescharacteristic, complemented, Sylow
411C2—161no—
412C2#7185211yescharacteristic
413C2#7186311yescharacteristic
414C2#7187611yescharacteristic
415C2#7195211yescharacteristic
416C2#7195311yescharacteristic
417C2#7195611yescharacteristic
418C2#7195711yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2820C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#2820D3
hypocenterC3#10427
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2820

Complements

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

NH ≅ G/Ntype
2. #71741C2semidirect
3. C2×#4543C2semidirect
6. #69550C2semidirect
7. #67630C2semidirect
410. C3#10427semidirect

Maximal subgroups

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

subgroupindex
C3×#28202
#717412
C2×#45432
#717262
C2×#45402
#695502
#676302
#104273