Finite Group Databasebeta

← #10692848

Subgroups — #10692848

Order 768.

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

normal subgroups only

This group has 486 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#82971C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic, complemented
3C3×#1533C211yesmaximal, characteristic
4#71719C211yesmaximal, characteristic
5#70830C211yesmaximal, characteristic, complemented
6#67750C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9448—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#944C411yescharacteristic
11C2×#936C411yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#432C2211yescharacteristic
15#5229C411yescharacteristic
16#5221C411yescharacteristic
17#4948—21no—
18#4948—21no—
19#4944—21no—
20#4944—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4828—22no—
25#4719—21no—
26#4718—21no—
27#4701—22no—
28C3×#329—22no—
29#4540—22no—
30#4469—21no—
31#4469—21no—
32#2983—31no—
33#2545—31no—
34#1533—31no—
35#1533D311yescharacteristic
36#1467—31no—
37#1285—31no—
38#1285—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×Dic6C2×C411yescharacteristic
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53#944—42no—
54#936—42no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4D411yescharacteristic
60C2×Dic3⋊C4D411yescharacteristic
61C2×C4×Dic3C2×C411yescharacteristic
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#815—22no—
81Dic3⋊C8—22no—
82Dic3⋊C8—22no—
83C8×Dic3—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803D412yes—
88C2×C4⋊Q8—31no—
89C2×#124—31no—
90C2×#124C2×D311yescharacteristic
91C2×C4×Q8—31no—
92#446—62no—
93#446—62no—
94#432—31no—
95#432—31no—
96#432—61no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—61no—
100#396—61no—
101#396—61no—
102#391—31no—
103#390—31no—
104#388—61no—
105#388—61no—
106C4×C4⋊C4—31no—
107#329—22no—
108#329—62no—
109#324—61no—
110#324—61no—
111C24×C3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C22×C3×C4—21no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3C22⋊C411yescharacteristic
127C22×Dic3C22⋊C411yescharacteristic
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—22no—
133C2×Dic6—41no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#217—42no—
145#217—42no—
146#217—42no—
147#217—42no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—42no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—42no—
156Dic3⋊C4—42no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C22×Q8—31no—
166C22×Q8C4×D311yescharacteristic
167C22×D4C4×D311yescharacteristic
168C4⋊Q8—122no—
169#124—21no—
170#124—21no—
171#124—21no—
172#124—21no—
173#124—122no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—31no—
193C2×C42—31no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—62no—
198C4⋊C8—62no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#97—21no—
206#97—32no—
207#97—62no—
208#97—61no—
209#97#6712yes—
210C4×C8—62no—
211#95—61no—
212#95—61no—
213#95—61no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3C2×C22⋊C411yescharacteristic
222C3×Q8—81no—
223C3×Q8—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—22no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3#10012yes—
255Dic6—42no—
256Dic6—42no—
257C3×C8—42no—
258C3×C8—81no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C24—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—62no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—121no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—62no—
287C2×C8—121no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—122no—
295C4⋊C4—122no—
296C4⋊C4—122no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—122no—
302C22⋊C4—122no—
303C22⋊C4—122no—
304C22⋊C4—122no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—61no—
309C42—121no—
310C42—121no—
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3#38611yescharacteristic
325C22×C3C2×#10011yescharacteristic
326C22×C3#41311yescharacteristic
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—81no—
334Dic3—22no—
335Dic3—42no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—81no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23D3×C22⋊C411yescharacteristic
348Q8—81no—
349Q8—81no—
350Q8—122no—
351Q8—122no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—32no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—62no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4#67⋊C411yescharacteristic
378C2×C4#88311yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4C4×#6711yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93811yescharacteristic
383C8—42no—
384C8—81no—
385C8—122no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—161no—
390C2×C3#188411yescharacteristic
391C2×C3#190011yescharacteristic
392C2×C3#191211yescharacteristic
393C22—21no—
394C22—21no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22D3×#10011yescharacteristic
403C22#474511yescharacteristic
404C22#495611yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—62no—
412C4—81no—
413C4—122no—
414C4—121no—
415C4—121no—
416C4—121no—
417C4—241no—
418C3#944811yescharacteristic, complemented, Sylow
419C2—21no—
420C2—21no—
421C2—161no—
422C2#7008111yescharacteristic
423C2#7041111yescharacteristic
424C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1533C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1533D3
hypocenterC3#9448
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #79314C2semidirect
5. #70830C2semidirect
6. #67750C2semidirect
7. #67749C2semidirect
418. C3#9448semidirect

Maximal subgroups

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

subgroupindex
#829712
#793142
C3×#15332
#717192
#708302
#677502
#677492
#94483