Finite Group Databasebeta

← #10714016

Subgroups — #10714016

Order 768.

698 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D16C211yesmaximal, characteristic, complemented
2C2×#5151C211yesmaximal, characteristic, complemented
3C22×#826C211yesmaximal, characteristic, complemented
4#72038C211yesmaximal, characteristic, complemented
5C2×#4562C211yesmaximal, characteristic, complemented
6C2×#4509C211yesmaximal, characteristic, complemented
7#67804C211yesmaximal, characteristic, complemented
8#13325—31nomaximal, Sylow
9C22×C3×D8C2211yescharacteristic
10C22×C3×D8C2211yescharacteristic
11C2×#940—21no—
12C22×#213—21no—
13C22×D24C2211yescharacteristic
14C2×C3×D16—22no—
15C2×C3×D16—22no—
16C2×C3×D16—22no—
17C22×C3×C16C2211yescharacteristic
18#5221—21no—
19C2×#832—21no—
20#5151—22no—
21#5151—22no—
22C2×#826—22no—
23C2×#826—22no—
24C2×#826—22no—
25C2×#818C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C22×C3⋊C16C2211yescharacteristic
28#4562—22no—
29#4509—22no—
30C22×D16—31no—
31C22×D16D311yescharacteristic, complemented
32C2×#493—31no—
33C2×#357—31no—
34C2×#357—31no—
35#1970—31no—
36#1374—31no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×#67—42no—
40C22×D12—21no—
41C2×C3×D8—21no—
42C2×C3×D8—21no—
43C2×C3×D8—21no—
44C2×C3×D8—21no—
45C2×C3×D8—21no—
46C2×C3×D8—21no—
47C2×C3×D8—21no—
48C2×C3×D8—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8D412yes—
52C2×C3×D8D412yescomplemented
53C22×C3×C8C2311yescharacteristic
54C2×#217—42no—
55#940—24no—
56#940—42no—
57C2×#213—24no—
58C2×#213—42no—
59C2×#213—42no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C22×C3⋊C8D411yescharacteristic
63C2×D24—21no—
64C2×D24—21no—
65C2×D24—21no—
66C2×D24—21no—
67C2×D24—21no—
68C2×D24D412yes—
69C3×D16—28no—
70C2×C3×C16—21no—
71C2×C3×C16—21no—
72C2×C3×C16D412yescomplemented
73#832—42no—
74#826—28no—
75#818—21no—
76#818—21no—
77C8×Dic3—21no—
78C8×Dic3—21no—
79C2×C3⋊C16—21no—
80C2×C3⋊C16—21no—
81C2×C3⋊C16D412yescomplemented
82C22×D8—31no—
83C22×D8—61no—
84C22×D8C2×D311yescharacteristic
85C22×D8C2×D311yescharacteristic
86C2×#127—61no—
87C2×D16—22no—
88C2×D16—22no—
89C2×D16—22no—
90C2×D16—62no—
91C2×D16—62no—
92C2×D16—62no—
93C22×C16—31no—
94C22×C16C2×D311yescharacteristic
95#493—62no—
96#493—62no—
97C2×#107—31no—
98C2×#102—61no—
99C2×C4×C8—31no—
100#390—61no—
101#357—62no—
102#357—62no—
103C24×C3—41no—
104C24×C3—41no—
105C23×D3—41no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—48no—
120C2×#67—84no—
121C2×#67—84no—
122C22×Dic3—41no—
123C22×Dic3D812yes—
124C2×D12—22no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C3×D8—22no—
131C3×D8—22no—
132C3×D8—22no—
133C3×D8—22no—
134C3×D8—41no—
135C3×D8—41no—
136C3×D8—41no—
137C3×D8—41no—
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8#4111yescharacteristic
144#217—84no—
145#213—48no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8D812yes—
153D24—22no—
154D24—22no—
155D24—41no—
156D24—41no—
157C3×C16—22no—
158C3⋊C16—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—61no—
162C22×D4—122no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—32no—
174C2×D8—64no—
175C2×D8—61no—
176C2×D8—61no—
177C2×D8—61no—
178C2×D8—61no—
179C2×D8—61no—
180C2×D8—122no—
181C2×D8—122no—
182C2×D8#6712yes—
183C2×D8#6712yescomplemented
184C22×C8—31no—
185C22×C8C22×D311yescharacteristic
186#127—64no—
187#127—122no—
188C2×C4⋊C4—61no—
189C2×C22⋊C4—122no—
190C2×C42—31no—
191D16—28no—
192D16—68no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—32no—
196C2×C16—61no—
197C2×C16—61no—
198C2×C16#6712yescomplemented
199#107—61no—
200#107—61no—
201#102—122no—
202C4×C8—61no—
203C4×C8—61no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—82no—
211C23×C3—82no—
212C23×C3—82no—
213C23×C3—82no—
214C23×C3C2×D811yescharacteristic
215C22×D3—42no—
216C22×D3—81no—
217C22×D3—81no—
218C22×D3—82no—
219C22×D3—82no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234#67—816no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—81no—
238C2×Dic3—82no—
239C2×Dic3D1614yescomplemented
240D12—42no—
241D12—42no—
242D12—81no—
243D12—81no—
244C3×C8#12011yescharacteristic
245C3×C8#12011yescharacteristic
246C3×C8#12111yescharacteristic
247C3×C8#12111yescharacteristic
248C3⋊C8—22no—
249C24—41no—
250C24—41no—
251C24—121no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—62no—
265C2×D4—128no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—244no—
272C2×D4—244no—
273C22×C4—32no—
274C22×C4—121no—
275C22×C4D3×D411yescharacteristic
276D8—22no—
277D8—22no—
278D8—22no—
279D8—22no—
280D8—41no—
281D8—41no—
282D8—41no—
283D8—41no—
284D8—62no—
285D8—62no—
286D8—128no—
287D8—121no—
288D8—121no—
289C2×C8—32no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8D3×D411yescharacteristic
293C2×C8D3×D411yescharacteristic
294C2×C8#23711yescharacteristic
295C2×C8C2×#6711yescharacteristic
296C2×C8C2×#6711yescharacteristic
297C2×C8C2×#6711yescharacteristic
298C4⋊C4—121no—
299C4⋊C4—121no—
300C22⋊C4—244no—
301C42—61no—
302C42—61no—
303C16—22no—
304C16—62no—
305C22×C3—84no—
306C22×C3—84no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—162no—
310C22×C3—162no—
311C22×C3—161no—
312C22×C3—161no—
313C22×C3—161no—
314C22×C3—161no—
315C22×C3#44711yescharacteristic
316C22×C3#45911yescharacteristic
317C22×C3#49311yescharacteristic
318C22×C3C2×D1612yes—
319C22×C3#50912yes—
320C2×D3—84no—
321C2×D3—82no—
322C2×D3—162no—
323C2×D3—161no—
324C2×D3—161no—
325C3×C4#39211yescharacteristic
326C3×C4#44711yescharacteristic
327C3×C4#45911yescharacteristic
328C3×C4#49611yescharacteristic
329Dic3—24no—
330Dic3—161no—
331C23—42no—
332C23—42no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—122no—
342C23—241no—
343C23—241no—
344C23—242no—
345C23—242no—
346C23D3×D811yescharacteristic
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—122no—
356D4—122no—
357D4—2416no—
358D4—241no—
359D4—241no—
360C2×C4—34no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—241no—
364C2×C4—242no—
365C2×C4D3×D811yescharacteristic
366C2×C4#91111yescharacteristic
367C2×C4#91111yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93911yescharacteristic
370C2×C4#94011yescharacteristic
371C8—62no—
372C8#93711yescharacteristic
373C8#93811yescharacteristic
374C8#93911yescharacteristic
375C8#95311yescharacteristic
376C2×C3—162no—
377C2×C3—162no—
378C2×C3#201111yescharacteristic
379C2×C3#218412yes—
380C2×C3#220612yes—
381C2×C3#224611yescharacteristic
382C2×C3#225011yescharacteristic
383D3—162no—
384C22—84no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22—162no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—162no—
394C22—244no—
395C22—242no—
396C22—482no—
397C22—481no—
398C22—481no—
399C22D3×D1612yes—
400C22#491112yes—
401C22#515011yescharacteristic
402C22#515111yescharacteristic
403C22#515611yescharacteristic
404C4—64no—
405C4—481no—
406C4#515011yescharacteristic
407C4#515411yescharacteristic
408C4#515611yescharacteristic
409C4#522311yescharacteristic
410C3#1332511yescharacteristic, complemented, Sylow
411C2—162no—
412C2—162no—
413C2—482no—
414C2#7146212yes—
415C2#7146411yescharacteristic
416C2#7146611yescharacteristic
417C2#7146712yes—
418C2#7204411yescharacteristic
subgroupG/H
centerC23D3×D8
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC22×C3×D16C2
socleC23×C3C2×D8
hypercenterC22×D16D3
hypocenterC3#13325
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×D16

Complements

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

NH ≅ G/Ntype
1. C22×C3×D16C2semidirect
2. C2×#5151C2semidirect
3. C22×#826C2semidirect
4. #72038C2semidirect
5. C2×#4562C2semidirect
6. C2×#4509C2semidirect
7. #67804C2semidirect
31. C22×D16D3semidirect
52. C2×C3×D8D4semidirect
72. C2×C3×C16D4semidirect
81. C2×C3⋊C16D4semidirect
183. C2×D8#67semidirect
198. C2×C16#67semidirect
239. C2×Dic3D16semidirect
410. C3#13325semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D162
C2×#51512
C22×#8262
#720382
C2×#45622
C2×#45092
#678042
#133253