Finite Group Databasebeta

← #10714033

Subgroups — #10714033

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×SD32C211yesmaximal, characteristic, complemented
2C22×#828C211yesmaximal, characteristic, complemented
3C2×#5151C211yesmaximal, characteristic
4#72066C211yesmaximal, characteristic, complemented
5C2×#4562C211yesmaximal, characteristic, complemented
6C2×#4509C211yesmaximal, characteristic, complemented
7#67804C211yesmaximal, characteristic, complemented
8#13328—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C22×C3×D8C2211yescharacteristic
11C2×#940—21no—
12C22×#213—21no—
13C22×D24C2211yescharacteristic
14C2×C3×SD32—22no—
15C2×C3×SD32—22no—
16C2×C3×SD32—22no—
17C22×C3×C16C2211yescharacteristic
18#5229—21no—
19C2×#835—21no—
20C2×#828—22no—
21C2×#828—22no—
22C2×#828—22no—
23#5151—22no—
24#5151—22no—
25C2×#818C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C22×C3⋊C16C2211yescharacteristic
28#4562—22no—
29#4509—22no—
30C22×SD32—31no—
31C22×SD32D311yescharacteristic, complemented
32C2×#493—31no—
33C2×#357—31no—
34C2×#357—31no—
35#1969—31no—
36#1374—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×#67—42no—
40C22×D12—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16—21no—
46C2×C3×Q16D412yescomplemented
47C2×C3×D8—21no—
48C2×C3×D8—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8D412yes—
53C22×C3×C8C2311yescharacteristic
54#940—24no—
55#940—42no—
56C2×#213—24no—
57C2×#213—42no—
58C2×#213—42no—
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3D411yescharacteristic
61C22×C3⋊C8D411yescharacteristic
62C2×D24—21no—
63C2×D24—21no—
64C2×D24—21no—
65C2×D24—21no—
66C2×D24—21no—
67C2×D24D412yes—
68C3×SD32—28no—
69C2×C3×C16—21no—
70C2×C3×C16—21no—
71C2×C3×C16D412yescomplemented
72#835—42no—
73C3⋊#95—42no—
74#828—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×Q16C2×D311yescharacteristic
83C22×D8—31no—
84C22×D8—61no—
85C22×D8C2×D311yescharacteristic
86C2×#127—61no—
87C2×SD32—22no—
88C2×SD32—22no—
89C2×SD32—22no—
90C2×SD32—62no—
91C2×SD32—62no—
92C2×SD32—62no—
93C22×C16—31no—
94C22×C16C2×D311yescharacteristic
95#493—62no—
96#493—62no—
97C2×#107—31no—
98C2×#103—61no—
99C2×C4×C8—31no—
100#391—61no—
101#357—62no—
102#357—62no—
103C24×C3—41no—
104C23×D3—41no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×D4—22no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C22×C3×C4—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×Q16—22no—
131C3×Q16—22no—
132C3×Q16—41no—
133C3×Q16—41no—
134C3×D8—22no—
135C3×D8—22no—
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#213—48no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8D812yes—
152D24—22no—
153D24—22no—
154D24—41no—
155D24—41no—
156C3×C16—22no—
157C3⋊C16—22no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—61no—
161C22×D4—122no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16#6712yescomplemented
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—
183C22×C8—31no—
184C22×C8C22×D311yescharacteristic
185#127—64no—
186#127—122no—
187C2×C4⋊C4—61no—
188C2×C42—31no—
189SD32—28no—
190SD32—68no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—32no—
194C2×C16—61no—
195C2×C16—61no—
196C2×C16#6712yescomplemented
197#107—61no—
198#107—61no—
199#103—122no—
200C4×C8—61no—
201C4×C8—61no—
202#95—122no—
203C23×C3—42no—
204C23×C3—81no—
205C23×C3—81no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3C2×D811yescharacteristic
209C22×D3—42no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—82no—
213C22×D3—82no—
214C3×Q8—42no—
215C3×Q8—42no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—81no—
221C3×D4—81no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231#67—816no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—81no—
235C2×Dic3—82no—
236C2×Dic3SD3214yescomplemented
237D12—42no—
238D12—42no—
239D12—81no—
240D12—81no—
241C3×C8#12011yescharacteristic
242C3×C8#12011yescharacteristic
243C3×C8#12111yescharacteristic
244C3×C8#12111yescharacteristic
245C3⋊C8—22no—
246C24—41no—
247C24—121no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—62no—
261C2×D4—128no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—244no—
268C2×D4—244no—
269C22×C4—32no—
270C22×C4—41no—
271C22×C4—121no—
272C22×C4D3×D411yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—41no—
276Q16—41no—
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281D8—62no—
282D8—62no—
283D8—128no—
284D8—121no—
285D8—121no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8D3×D411yescharacteristic
290C2×C8D3×D411yescharacteristic
291C2×C8#23711yescharacteristic
292C2×C8C2×#6711yescharacteristic
293C2×C8C2×#6711yescharacteristic
294C2×C8C2×#6711yescharacteristic
295C4⋊C4—121no—
296C4⋊C4—121no—
297C42—61no—
298C42—61no—
299C16—22no—
300C16—62no—
301C22×C3—84no—
302C22×C3—82no—
303C22×C3—162no—
304C22×C3—161no—
305C22×C3—161no—
306C22×C3#44711yescharacteristic
307C22×C3#45911yescharacteristic
308C22×C3#49311yescharacteristic
309C22×C3C2×SD3212yes—
310C22×C3#50912yes—
311C2×D3—84no—
312C2×D3—82no—
313C2×D3—162no—
314C2×D3—161no—
315C2×D3—161no—
316C3×C4—82no—
317C3×C4#39211yescharacteristic
318C3×C4#44711yescharacteristic
319C3×C4#45911yescharacteristic
320C3×C4#49611yescharacteristic
321Dic3—24no—
322Dic3—161no—
323C23—42no—
324C23—81no—
325C23—81no—
326C23—82no—
327C23—82no—
328C23—122no—
329C23—241no—
330C23—241no—
331C23—242no—
332C23—242no—
333C23D3×D811yescharacteristic
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—122no—
343D4—122no—
344D4—2416no—
345D4—241no—
346D4—241no—
347C2×C4—34no—
348C2×C4—42no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—241no—
354C2×C4—242no—
355C2×C4D3×D811yescharacteristic
356C2×C4#91111yescharacteristic
357C2×C4#91111yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#93911yescharacteristic
360C2×C4#94011yescharacteristic
361C8—62no—
362C8#93711yescharacteristic
363C8#93811yescharacteristic
364C8#93911yescharacteristic
365C8#95311yescharacteristic
366C2×C3—162no—
367C2×C3#201111yescharacteristic
368C2×C3#218712yes—
369C2×C3#220712yes—
370C2×C3#224811yescharacteristic
371C2×C3#225011yescharacteristic
372D3—162no—
373C22—84no—
374C22—82no—
375C22—162no—
376C22—161no—
377C22—161no—
378C22—244no—
379C22—242no—
380C22—482no—
381C22—481no—
382C22—481no—
383C22D3×SD3212yes—
384C22#491412yes—
385C22#515011yescharacteristic
386C22#515111yescharacteristic
387C22#515611yescharacteristic
388C4—64no—
389C4—82no—
390C4—481no—
391C4#515011yescharacteristic
392C4#515411yescharacteristic
393C4#515611yescharacteristic
394C4#522311yescharacteristic
395C3#1332811yescharacteristic, complemented, Sylow
396C2—162no—
397C2—482no—
398C2#7147512yes—
399C2#7147911yescharacteristic
400C2#7148111yescharacteristic
401C2#7148512yes—
402C2#7204411yescharacteristic
subgroupG/H
centerC23D3×D8
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC22×C3×SD32C2
socleC23×C3C2×D8
prime radicalC2×#5151C2
hypercenterC22×SD32D3
hypocenterC3#13328
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C22×C3×SD32C2semidirect
2. C22×#828C2semidirect
4. #72066C2semidirect
5. C2×#4562C2semidirect
6. C2×#4509C2semidirect
7. #67804C2semidirect
31. C22×SD32D3semidirect
46. C2×C3×Q16D4semidirect
71. C2×C3×C16D4semidirect
81. C2×C3⋊C16D4semidirect
167. C2×Q16#67semidirect
196. C2×C16#67semidirect
236. C2×Dic3SD32semidirect
395. C3#13328semidirect

Maximal subgroups

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

subgroupindex
C22×C3×SD322
C22×#8282
C2×#51512
#720662
C2×#45622
C2×#45092
#678042
#133283