Finite Group Databasebeta

← #10714535

Subgroups — #10714535

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5191C211yesmaximal, characteristic
2C2×C3×#358C211yesmaximal, characteristic, complemented
3C2×#4564C211yesmaximal, characteristic, complemented
4C2×#4509C211yesmaximal, characteristic
5#70612C211yesmaximal, characteristic, complemented
6C2×#4491C211yesmaximal, characteristic
7#67804C211yesmaximal, characteristic, complemented
8#13408—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×#947—21no—
11C22×#215—21no—
12C22×D24C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C2×C3×#107C2211yescharacteristic
15#5191—22no—
16#5191—22no—
17C2×#818C2211yescharacteristic
18C2×C8×Dic3C2211yescharacteristic
19C22×C3⋊C16C2211yescharacteristic
20#4984—21no—
21C3×#358—22no—
22#4564—22no—
23#4529—21no—
24#4509—22no—
25#4491—22no—
26C2×#495—31no—
27C2×#358—31no—
28C2×#358D311yescharacteristic, complemented
29C2×#357—31no—
30C2×#357—31no—
31#1942—31no—
32#1374—31no—
33C22×C3×Q8—21no—
34C22×D12—21no—
35C2×C3×Q16—21no—
36C2×C3×Q16—21no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16D412yes—
41C22×C3×C8C2311yescharacteristic
42C2×C3×C4⋊C4—21no—
43#947—24no—
44#947—42no—
45C2×#215—24no—
46C2×#215—42no—
47C2×#215—42no—
48C2×(C2×D3)⋊C4—41no—
49C2×(C2×D3)⋊C4—41no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—41no—
52C2×C4×Dic3D411yescharacteristic
53C22×C3⋊C8D411yescharacteristic
54C2×D24—21no—
55C2×D24—21no—
56C2×D24—21no—
57C2×D24—21no—
58C2×D24—21no—
59C2×D24D412yes—
60C2×C3×C16—21no—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C3×#107—21no—
64C3×#107D412yes—
65C3⋊#95—41no—
66#818—21no—
67#818—21no—
68C8×Dic3—21no—
69C8×Dic3—21no—
70C2×C3⋊C16—21no—
71C2×C3⋊C16—21no—
72C2×C3⋊C16—21no—
73C22×Q16C2×D311yescharacteristic
74C22×SD16—61no—
75C22×D8—31no—
76C2×#124—61no—
77C22×C16—31no—
78C22×C16C2×D311yescharacteristic
79#495—62no—
80#495—62no—
81C2×#107—31no—
82C2×#107C2×D311yescharacteristic
83C2×C4×C8—31no—
84#389—61no—
85#358—22no—
86#358—62no—
87#357—62no—
88#357—62no—
89#340—61no—
90C23×D3—41no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—41no—
101C22×Dic3D811yescharacteristic
102C22×Dic3D811yescharacteristic
103C2×D12—22no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—41no—
108C2×D12—41no—
109C3×Q16—22no—
110C3×Q16—22no—
111C3×Q16—41no—
112C3×Q16—41no—
113C2×C3×C8C2×D411yescharacteristic
114C2×C3×C8C2×D411yescharacteristic
115C2×C3×C8C2×D411yescharacteristic
116C2×C3×C8#4111yescharacteristic
117C2×C3×C8#4111yescharacteristic
118C2×C3×C8#4111yescharacteristic
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121#215—48no—
122(C2×D3)⋊C4—44no—
123(C2×D3)⋊C4—44no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—82no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8SD1612yes—
132D24—22no—
133D24—22no—
134D24—41no—
135D24—41no—
136C3×C16—41no—
137C3⋊C16—41no—
138C22×Q8—21no—
139C22×D4—61no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16D1212yes—
146C2×SD16—64no—
147C2×SD16—122no—
148C2×SD16—122no—
149C2×D8—32no—
150C2×D8—61no—
151C2×D8—61no—
152C2×D8—61no—
153C2×D8—61no—
154C2×D8—61no—
155C22×C8—31no—
156C22×C8C22×D311yescharacteristic
157#124—64no—
158#124—122no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—121no—
162C2×C22⋊C4—121no—
163C2×C22⋊C4—121no—
164C2×C42—31no—
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—61no—
169C2×C16—61no—
170C2×C16—61no—
171#107—21no—
172#107—61no—
173#107—61no—
174#107#6712yes—
175C4×C8—61no—
176C4×C8—61no—
177#95—121no—
178C23×C3C2×D811yescharacteristic
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C22×D3—81no—
183C22×D3—81no—
184C22×D3—81no—
185C22×D3—81no—
186C3×Q8—42no—
187C3×Q8—42no—
188C3×Q8—81no—
189C3×Q8—81no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×C3×C4C2×SD1611yescharacteristic
200C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211D12—42no—
212D12—42no—
213D12—81no—
214D12—81no—
215C3×C8#12111yescharacteristic
216C3×C8#12111yescharacteristic
217C3×C8#12311yescharacteristic
218C3×C8#12311yescharacteristic
219C3⋊C8—22no—
220C24—121no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×D4—62no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—121no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—121no—
238C22×C4D3×D411yescharacteristic
239Q16—22no—
240Q16—22no—
241Q16—41no—
242Q16—41no—
243SD16—128no—
244D8—62no—
245D8—62no—
246D8—121no—
247D8—121no—
248C2×C8—32no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8C2×D1211yescharacteristic
252C2×C8#23511yescharacteristic
253C2×C8D3×D411yescharacteristic
254C2×C8#23711yescharacteristic
255C2×C8#23711yescharacteristic
256C2×C8C2×#6711yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—242no—
262C22⋊C4—124no—
263C22⋊C4—124no—
264C42—61no—
265C42—61no—
266C16—41no—
267C16—121no—
268C22×C3#44711yescharacteristic
269C22×C3#45911yescharacteristic
270C22×C3#48611yescharacteristic
271C22×C3#50811yescharacteristic
272C22×C3#50811yescharacteristic
273C22×C3#50911yescharacteristic
274C22×C3#51011yescharacteristic
275C2×D3—82no—
276C2×D3—82no—
277C2×D3—82no—
278C2×D3—161no—
279C2×D3—161no—
280C2×D3—161no—
281C2×D3—161no—
282C3×C4—82no—
283C3×C4—82no—
284C3×C4#39411yescharacteristic
285C3×C4#45011yescharacteristic
286C3×C4#46011yescharacteristic
287C3×C4#49011yescharacteristic
288Dic3—41no—
289Dic3—41no—
290Dic3—161no—
291C23—122no—
292C23—241no—
293C23—241no—
294C23—241no—
295C23—241no—
296C23—241no—
297C23—241no—
298C23D3×D811yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—81no—
302Q8—81no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—241no—
320C2×C4—241no—
321C2×C4—241no—
322C2×C4#88211yescharacteristic
323C2×C4#88511yescharacteristic
324C2×C4#91111yescharacteristic
325C2×C4D3×SD1611yescharacteristic
326C2×C4#91411yescharacteristic
327C2×C4#93911yescharacteristic
328C8—62no—
329C8#88311yescharacteristic
330C8#88611yescharacteristic
331C8#93011yescharacteristic
332C8#93811yescharacteristic
333C2×C3#204111yescharacteristic
334C2×C3#218611yescharacteristic
335C2×C3#218911yescharacteristic
336C2×C3#221011yescharacteristic
337C2×C3#221111yescharacteristic
338C2×C3#224211yescharacteristic
339C2×C3#224311yescharacteristic
340D3—162no—
341C22—242no—
342C22—242no—
343C22—242no—
344C22—481no—
345C22—481no—
346C22—481no—
347C22—481no—
348C22#476011yescharacteristic
349C22#477311yescharacteristic
350C22#491411yescharacteristic
351C22#491611yescharacteristic
352C22#491811yescharacteristic
353C22#491911yescharacteristic
354C22#515011yescharacteristic
355C4—82no—
356C4—82no—
357C4—121no—
358C4—121no—
359C4—481no—
360C4#476411yescharacteristic
361C4#477611yescharacteristic
362C4#496111yescharacteristic
363C4#516311yescharacteristic
364C3#1340811yescharacteristic, complemented, Sylow
365C2—482no—
366C2#6921611yescharacteristic
367C2#6921711yescharacteristic
368C2#6922411yescharacteristic
369C2#6922611yescharacteristic
370C2#7017711yescharacteristic
371C2#7147711yescharacteristic
372C2#7149811yescharacteristic
subgroupG/H
centerC23D3×D8
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#358C2
socleC23×C3C2×D8
prime radicalC22×D24C22
hypercenterC2×#358D3
hypocenterC3#13408
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×C3×#358C2semidirect
3. C2×#4564C2semidirect
5. #70612C2semidirect
7. #67804C2semidirect
28. C2×#358D3semidirect
364. C3#13408semidirect

Maximal subgroups

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

subgroupindex
C2×#51912
C2×C3×#3582
C2×#45642
C2×#45092
#706122
C2×#44912
#678042
#134083