Finite Group Databasebeta

← #10698664

Subgroups — #10698664

Order 768.

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

normal subgroups only

This group has 444 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#69548C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#10426—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—
20#4935C2211yescharacteristic
21C2×#803C2211yescharacteristic
22C2×#803C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4543—22no—
25#4540—22no—
26#4529—21no—
27#4529—21no—
28#4529—21no—
29#4529—21no—
30#2820D311yescharacteristic
31#1992—31no—
32#1959—31no—
33#1836—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#803—21no—
75#803—21no—
76#803—21no—
77#803—21no—
78#803—21no—
79#803D412yes—
80C2×#125C2×D311yescharacteristic
81C2×#124C2×D311yescharacteristic
82C2×#103—61no—
83C2×#102—61no—
84C2×#97—31no—
85C2×#97—31no—
86C2×#97—31no—
87#399—21no—
88#397—21no—
89#396—21no—
90#396—21no—
91#395—21no—
92#394—21no—
93#384—31no—
94#382C2×D311yescharacteristic
95#340—61no—
96#340—61no—
97#340—61no—
98#340—61no—
99#330—62no—
100#329—62no—
101C24×C3—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—81no—
130C3×C4⋊C4—81no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—81no—
136C3×C22⋊C4—81no—
137C3×C22⋊C4—81no—
138C3×C22⋊C4—81no—
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142C3×C42#4111yescharacteristic
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—42no—
146C4⋊Dic3—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C22×Q8#6711yescharacteristic
157C22×D4#6711yescharacteristic
158C22×C8—61no—
159C22×C8—61no—
160C22×C8—61no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C4⋊C4#6711yescharacteristic
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42C22×D311yescharacteristic
187#103—122no—
188#102—122no—
189#97—32no—
190#97—61no—
191#97—61no—
192#97—61no—
193#97—61no—
194#97—61no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C3×D4—42no—
210C3×D4—81no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
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#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4#12411yescharacteristic
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244C24—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—121no—
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—121no—
266C2×C8—121no—
267C2×C8—121no—
268C2×C8—121no—
269C2×C8—121no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—122no—
286C4⋊C4—122no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42D3×D411yescharacteristic
296C42#23711yescharacteristic
297C42#23711yescharacteristic
298C42#23711yescharacteristic
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3#39411yescharacteristic
307C22×C3#45311yescharacteristic
308C22×C3#45411yescharacteristic
309C22×C3#45411yescharacteristic
310C22×C3#45511yescharacteristic
311C22×C3#45511yescharacteristic
312C22×C3#45611yescharacteristic
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—81no—
324Dic3—161no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#95311yescharacteristic
333Q8—42no—
334Q8—81no—
335D4—42no—
336D4—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
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—241no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4#93511yescharacteristic
362C2×C4#93511yescharacteristic
363C2×C4#93611yescharacteristic
364C2×C4#93811yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#93911yescharacteristic
367C8—122no—
368C8—122no—
369C8—122no—
370C2×C3—161no—
371C2×C3#165711yescharacteristic
372C2×C3#166111yescharacteristic
373C2×C3#166211yescharacteristic
374C2×C3#166411yescharacteristic
375C2×C3#203711yescharacteristic
376C2×C3#203811yescharacteristic
377C2×C3#204011yescharacteristic
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#522211yescharacteristic
386C22#524111yescharacteristic
387C22#524211yescharacteristic
388C22#524211yescharacteristic
389C22#524511yescharacteristic
390C22#524511yescharacteristic
391C22#524611yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—481no—
404C3#1042611yescharacteristic, complemented, Sylow
405C2—161no—
406C2#7185211yescharacteristic
407C2#7186311yescharacteristic
408C2#7187711yescharacteristic
409C2#7195311yescharacteristic
410C2#7195411yescharacteristic
411C2#7195511yescharacteristic
412C2#7195711yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2820C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#2820D3
hypocenterC3#10426
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. #69548C2semidirect
7. #67630C2semidirect
404. C3#10426semidirect

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
#695482
#676302
#104263