Finite Group Databasebeta

← #10697843

Subgroups — #10697843

Order 768.

678 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2858C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#83166C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#10284—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×C3×#124—22no—
12#5477C2211yescharacteristic
13C3×#390—22no—
14C3×#390—22no—
15C3×C43C2211yescharacteristic
16#5217—22no—
17#5088—21no—
18#5087—22no—
19#5078—21no—
20#5073—21no—
21#5072—22no—
22#5070—21no—
23#4967—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4544—22no—
30#3226—31no—
31#3205—31no—
32#2858D311yescharacteristic, complemented
33#1962—31no—
34#1546—31no—
35#1545—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D412yes—
41C2×#235—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#124—44no—
51C3×#124—44no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4—28no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×C3×C42D412yes—
58C2×C3×C42C2311yescharacteristic
59C2×#213—24no—
60#930—21no—
61#930—21no—
62C2×#208—21no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68#832—24no—
69#830—21no—
70#830—21no—
71#809—24no—
72#807—24no—
73#805—44no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80#532—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#127C2×D311yescharacteristic
83C2×#124—22no—
84#500—61no—
85#493—61no—
86#486—61no—
87#486—61no—
88#480—62no—
89#474—62no—
90#459—62no—
91#447—62no—
92#431—31no—
93#431—31no—
94#423—31no—
95C2×#104—62no—
96#390—22no—
97#390—22no—
98C43C2×D311yescharacteristic
99#332—62no—
100#331—62no—
101C24×C3—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—44no—
109C2×C3×D4—44no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122#235—42no—
123#235—42no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—416no—
134C3×C42—22no—
135C3×C42—42no—
136C3×C42—42no—
137C3×C42D814yescomplemented
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#213—84no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147#208—42no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×#41—61no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4#6712yes—
156C2×D8—64no—
157C2×M16—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163#127—21no—
164#127—21no—
165#127—21no—
166#127—21no—
167#127—31no—
168#124—44no—
169#124—44no—
170#124—61no—
171#121—61no—
172#121—61no—
173C4×D4—61no—
174C4×D4—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6712yes—
177C2×C22⋊C4—28no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42#6712yes—
181C2×C42C22×D311yescharacteristic
182#107—64no—
183C4⋊C8—32no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187#104—124no—
188#102—64no—
189#102—64no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192C4×C8—61no—
193C4×C8—61no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3—42no—
197C23×C3—44no—
198C23×C3#12011yescharacteristic
199C22×D3—41no—
200C3×Q8—82no—
201C3×Q8—81no—
202C3×Q8—81no—
203C3×D4—44no—
204C3×D4—82no—
205C3×D4—82no—
206C3×D4—82no—
207C3×D4—82no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×C3×C4C2×D812yes—
224C2×C3×C4AGL(1,ℤ/8)12yes—
225#67—82no—
226C2×Dic3—41no—
227D12—42no—
228D12—42no—
229D12—42no—
230C4×D3—81no—
231C4×D3—81no—
232Dic6—42no—
233C3⋊C8—42no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—22no—
237#41—122no—
238#41—122no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—61no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—121no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—121no—
261C22×C4C2×#6711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264D8—244no—
265M16—122no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—121no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C22⋊C4—416no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—22no—
279C42—42no—
280C42—42no—
281C42#21314yescomplemented
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42#23711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—88no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3#39211yescharacteristic
293C22×C3#44712yes—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—44no—
300C3×C4—82no—
301C3×C4#45412yes—
302C3×C4#45512yes—
303C3×C4#45912yes—
304C3×C4#49311yescharacteristic
305C3×C4#49611yescharacteristic
306Dic3—81no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—42no—
311C23—121no—
312C23#95311yescharacteristic
313Q8—82no—
314Q8—81no—
315Q8—81no—
316Q8—122no—
317D4—44no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4—82no—
322D4—81no—
323D4—81no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—242no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—121no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4C2×#21312yes—
339C2×C4#93212yes—
340C2×C4#93711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C2×C4#93911yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#94011yescharacteristic
346C8—122no—
347C8—122no—
348C8—241no—
349C2×C3—22no—
350C2×C3—162no—
351C2×C3#200311yescharacteristic
352C2×C3#200411yescharacteristic
353C2×C3#201111yescharacteristic
354D3—161no—
355C22—21no—
356C22—21no—
357C22—88no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—241no—
362C22—241no—
363C22—241no—
364C22#521712yes—
365C22#522311yescharacteristic
366C4—21no—
367C4—21no—
368C4—44no—
369C4—82no—
370C4—241no—
371C4#506112yes—
372C4#506212yes—
373C4#507212yes—
374C4#507311yescharacteristic
375C4#507411yescharacteristic
376C3#1028411yescharacteristic, complemented, Sylow
377C2—22no—
378C2—162no—
379C2—481no—
380C2#7173011yescharacteristic
381C2#7173111yescharacteristic
382C2#7176711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2858C2
socleC22×C3#392
prime radical#83166C2
hypercenter#2858D3
hypocenterC3#10284
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2858

Complements

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

NH ≅ G/Ntype
1. C3×#2858C2semidirect
2. #83215C2semidirect
4. #71773C2semidirect
5. #71759C2semidirect
6. #69395C2semidirect
7. #67611C2semidirect
32. #2858D3semidirect
137. C3×C42D8semidirect
281. C42#213semidirect
376. C3#10284semidirect

Maximal subgroups

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

subgroupindex
C3×#28582
#832152
#831662
#717732
#717592
#693952
#676112
#102843