Finite Group Databasebeta

← #10696068

Subgroups — #10696068

Order 768.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3142C211yesmaximal, characteristic, complemented
2#83300C211yesmaximal, characteristic
3#83066C211yesmaximal, characteristic, complemented
4#71813C211yesmaximal, characteristic, complemented
5#71736C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#10019—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#491—21no—
13C3×#490—21no—
14C3×#488—22no—
15C3×#417—21no—
16C3×#417—21no—
17C3×C4×M16C2211yescharacteristic
18#5084—21no—
19#5072—21no—
20#5036—21no—
21#5034—21no—
22#5014—22no—
23#4993—21no—
24#4992—21no—
25#4991—21no—
26#4989—22no—
27#4981—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4543—22no—
32#4537—22no—
33#4459—22no—
34#3142D311yescharacteristic, complemented
35#3109—31no—
36#2633—31no—
37#1534—31no—
38#1514—31no—
39#1490—31no—
40#1288—31no—
41C2×C3×#41—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×#125—21no—
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×#213—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—22no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C3×#103—22no—
72C3×#103—22no—
73C3×#102—22no—
74C3×#102—22no—
75C3×#97—21no—
76C3×C4×C8D412yes—
77#832—22no—
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#809—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803D412yes—
90#532C2×D311yescharacteristic
91C2×#125C2×D311yescharacteristic
92C2×C4×D4—31no—
93#491—21no—
94#490—21no—
95#488—22no—
96#481—61no—
97#476—61no—
98#459—61no—
99#447—61no—
100#442—62no—
101#432—31no—
102#432—31no—
103#417—21no—
104#417—21no—
105C4×M16C2×D311yescharacteristic
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#327—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C3×#41—44no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—82no—
136C3×M16—42no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#213—44no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—82no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—82no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×#41—21no—
165C22×D4—31no—
166C23×C4—61no—
167C2×D8—62no—
168C2×M16—21no—
169C4⋊Q8#6711yescharacteristic
170#127#6711yescharacteristic
171#125—21no—
172#125—21no—
173#125—21no—
174#125—21no—
175#124—21no—
176#121—21no—
177#121—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4#6711yescharacteristic
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—62no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C42C22×D311yescharacteristic
194#106—62no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—22no—
198#103—22no—
199#102—22no—
200#102—22no—
201#102—62no—
202#102—62no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#97—21no—
206#97—32no—
207#97—61no—
208C4×C8#6712yes—
209#95—32no—
210#95—62no—
211#95—61no—
212#95—61no—
213C23×C3—41no—
214C23×C3#12011yescharacteristic
215C22×D3—22no—
216C22×D3—42no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—42no—
220C3×Q8—42no—
221C3×D4—42no—
222C3×D4—42no—
223C3×D4—81no—
224C3×D4—82no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—82no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246D12—22no—
247D12—22no—
248D12—41no—
249D12—41no—
250C4×D3—82no—
251C4×D3—82no—
252C3×C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—61no—
256#41—44no—
257C2×Q8—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—122no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—242no—
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4C2×#6711yescharacteristic
276D8—124no—
277M16—42no—
278C2×C8—22no—
279C2×C8—62no—
280C2×C8—121no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—242no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—124no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—242no—
300C42D3×D411yescharacteristic
301C42#23711yescharacteristic
302C42C2×#6711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3#39211yescharacteristic
310C22×C3#45412yes—
311C2×D3—44no—
312C2×D3—42no—
313C2×D3—82no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4—81no—
325Dic3—82no—
326Dic3—81no—
327C23—41no—
328C23—62no—
329C23—122no—
330C23—121no—
331C23—121no—
332C23—122no—
333C23#93711yescharacteristic
334Q8—42no—
335D4—42no—
336D4—42no—
337D4—62no—
338D4—62no—
339D4—81no—
340D4—82no—
341D4—121no—
342D4—121no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—82no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—242no—
359C2×C4—242no—
360C2×C4#93711yescharacteristic
361C2×C4#93811yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#94011yescharacteristic
365C2×C4#95311yescharacteristic
366C8—42no—
367C8—122no—
368C8—241no—
369C2×C3—22no—
370C2×C3—161no—
371C2×C3#200311yescharacteristic
372C2×C3#200811yescharacteristic
373C2×C3#201011yescharacteristic
374D3—82no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—124no—
381C22—122no—
382C22—242no—
383C22—241no—
384C22—241no—
385C22#506212yes—
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—242no—
397C4—241no—
398C3#1001911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—161no—
401C2—242no—
402C2#7173011yescharacteristic
403C2#7211811yescharacteristic
404C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3142C2
socleC22×C3#392
prime radical#83300C2
hypercenter#3142D3
hypocenterC3#10019
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. C3×#3142C2semidirect
3. #83066C2semidirect
4. #71813C2semidirect
5. #71736C2semidirect
6. #69855C2semidirect
7. #67666C2semidirect
34. #3142D3semidirect
398. C3#10019semidirect

Maximal subgroups

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

subgroupindex
C3×#31422
#833002
#830662
#718132
#717362
#698552
#676662
#100193