Finite Group Databasebeta

← #10824910

Subgroups — #10824910

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5184C211yesmaximal, characteristic, complemented
2#79071C211yesmaximal, characteristic
3#78946C212yesmaximal
4C3×#2047C211yesmaximal, characteristic, complemented
5#72074C211yesmaximal, characteristic
6#70533C211yesmaximal, characteristic, complemented
7#70337C212yesmaximal, complemented
8#70126C212yesmaximal
9Dic3×#101C212yesmaximal, complemented
10#68922C211yesmaximal, characteristic
11#68917C211yesmaximal, characteristic, complemented
12#23720—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C2×#942C2212yes—
16#5522C2211yescharacteristic
17C3×#424C2211yescharacteristic
18C3×#419C2212yes—
19C2×C3×#101C2212yes—
20C2×#836C2212yes—
21C2×#835C2212yes—
22#5184—22no—
23#5184—22no—
24#5133C2212yes—
25#5129C2211yescharacteristic
26C2×#822C2212yes—
27Dic3×M16C2212yes—
28Dic3×M16C2211yescharacteristic
29#5110C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#815C2211yescharacteristic
32#4999C2211yescharacteristic
33#4995C2212yes—
34#4840—22no—
35#4835—22no—
36#4817—22no—
37#4790—22no—
38#4789—22no—
39#4788—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×#137—32no—
45C2×#441—31no—
46#2923—31no—
47#2090—31no—
48#2079—31no—
49#2064—32no—
50#2052—32no—
51#2047D311yescharacteristic, complemented
52#2020—31no—
53#1778—31no—
54C4×#101—32no—
55C22×C3×Q8C2312yes—
56C2×#235C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C2×C3×M16C2312yes—
60C2×C3×M16C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#117C2311yescharacteristic
63Q8×Dic3—22no—
64Q8×Dic3—22no—
65Q8×Dic3—22no—
66Q8×Dic3—22no—
67C2×#216—22no—
68#942—24no—
69#942—24no—
70C2×#215—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4×Dic3—22no—
74C2×C4×Dic3C2311yescharacteristic
75C2×#208C2311yescharacteristic
76C2×#208C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78#893D412yes—
79#887D412yes—
80#67⋊C4—22no—
81#872—21no—
82C42×D3—21no—
83C3×C4⋊C8—22no—
84C3×#103—22no—
85C3×#103—22no—
86C3×#101—24no—
87#836—24no—
88#835—24no—
89#822—24no—
90#821—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93#817—22no—
94#816—22no—
95#815—22no—
96#815—21no—
97#815D412yes—
98#815D412yes—
99C8×Dic3—22no—
100C8×Dic3—21no—
101#808—22no—
102#807—22no—
103C4⋊(C3⋊C8)—22no—
104C2×#137—32no—
105C22×Q16C2×D311yescharacteristic
106C4×#41—31no—
107C2×C4×Q8—32no—
108#441—62no—
109#441—62no—
110#441—62no—
111#440—62no—
112C4×Q16—62no—
113C4×SD16—62no—
114#433—62no—
115#432—62no—
116C2×#108—32no—
117#428—32no—
118C2×#106—31no—
119#424—31no—
120#424C2×D311yescharacteristic
121#419C2×D312yes—
122#417—32no—
123C2×#103—32no—
124C2×#101—32no—
125C2×#101C2×D312yes—
126#408—31no—
127#408—31no—
128C4×M16—31no—
129C4×M16—32no—
130C2×#97—31no—
131#343—32no—
132#343—32no—
133#341—31no—
134#341C2×D311yescharacteristic
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8#4114yes—
141C22×C3×C4—22no—
142C22×C3×C4C2411yescharacteristic
143C2×#67—21no—
144C22×Dic3—22no—
145C22×Dic3#4112yes—
146#235—22no—
147#235—22no—
148C2×D12C2×D411yescharacteristic
149C2×C4×D3—21no—
150C2×C4×D3—22no—
151C2×Dic6C2×D411yescharacteristic
152C3×Q16—28no—
153C3×M16—22no—
154C3×M16—22no—
155C3×M16—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—21no—
159C2×C3×C8C2×D412yes—
160C3×C4⋊C4C2×D412yes—
161C3×C22⋊C4—21no—
162C3×C42—21no—
163#216—28no—
164#215—28no—
165(C2×D3)⋊C4—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—42no—
169C4⋊Dic3—42no—
170Dic3⋊C4—22no—
171C4×Dic3—22no—
172C4×Dic3—42no—
173C4×Dic3—42no—
174C4×Dic3C2×D412yes—
175C4×Dic3C2×D411yescharacteristic
176C4×Dic3C2×D411yescharacteristic
177#208—22no—
178#208—22no—
179#208#4114yes—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8C2×D412yes—
184C2×#41—31no—
185C22×Q8C22×D312yes—
186#137—64no—
187#137—64no—
188C2×Q16—24no—
189C2×Q16—22no—
190C2×Q16—62no—
191C2×SD16—62no—
192C2×M16—32no—
193C2×M16—31no—
194C2×M16C22×D312yes—
195C2×M16C22×D311yescharacteristic
196C22×C8—31no—
197C22×C8C22×D311yescharacteristic
198C4×Q8—32no—
199C4×Q8—62no—
200C4×Q8—62no—
201C4×Q8—62no—
202C4×Q8—62no—
203C4×D4—32no—
204C4×D4—62no—
205#117—61no—
206#117C22×D311yescharacteristic
207C2×C4⋊C4—32no—
208C2×C4⋊C4—62no—
209C2×C42—31no—
210C2×C42—61no—
211C2×C42—62no—
212#108—64no—
213#107—62no—
214#106—62no—
215#106—62no—
216C4⋊C8—22no—
217C4⋊C8—62no—
218#103—22no—
219#103—22no—
220#103—62no—
221#103—64no—
222#102—62no—
223#101—24no—
224#101—64no—
225C22⋊C8—62no—
226C22⋊C8—62no—
227#97—32no—
228#97—32no—
229#97—61no—
230#97—62no—
231C4×C8—61no—
232C4×C8—62no—
233C23×C3C2×#4111yescharacteristic
234C22×D3—21no—
235C3×Q8—24no—
236C3×Q8—24no—
237C3×Q8—42no—
238C3×Q8—42no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C22×D411yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×C3×C4C2×#4112yes—
247#67—42no—
248C2×Dic3—21no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—82no—
252C2×Dic3#12112yes—
253C2×Dic3#12112yes—
254D12—22no—
255C4×D3—41no—
256C4×D3—41no—
257C4×D3—82no—
258Dic6—22no—
259C3×C8—22no—
260C3×C8—41no—
261C3×C8—42no—
262C3⋊C8—24no—
263C3⋊C8—22no—
264C3⋊C8—41no—
265#41—62no—
266#41—62no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—31no—
273C2×Q8#23714yes—
274C2×D4—31no—
275C2×D4—61no—
276C22×C4—22no—
277C22×C4—32no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4C23×D311yescharacteristic
282Q16—28no—
283Q16—68no—
284SD16—68no—
285M16—22no—
286M16—22no—
287M16—22no—
288M16—34no—
289M16—62no—
290M16—62no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—32no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—61no—
298C2×C8D3×D412yes—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—122no—
303C4⋊C4—122no—
304C4⋊C4D3×D412yes—
305C22⋊C4—21no—
306C22⋊C4—62no—
307C42—21no—
308C42—31no—
309C42—32no—
310C42—31no—
311C42—62no—
312C42—122no—
313C42—122no—
314C22×C3—22no—
315C22×C3C2×#12111yescharacteristic
316C22×C3#52912yes—
317C2×D3—41no—
318C2×D3—42no—
319C3×C4—44no—
320C3×C4—42no—
321C3×C4#52511yescharacteristic
322C3×C4#52512yes—
323C3×C4#53211yescharacteristic
324Dic3—22no—
325Dic3—22no—
326Dic3—41no—
327Dic3—82no—
328C23—61no—
329C23D3×#4111yescharacteristic
330Q8—24no—
331Q8—24no—
332Q8—42no—
333Q8—42no—
334Q8—62no—
335D4—62no—
336D4—122no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—61no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—242no—
349C2×C4—242no—
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×D3×D411yescharacteristic
352C2×C4C2×#23712yes—
353C2×C4D3×#4112yes—
354C8—22no—
355C8—41no—
356C8—42no—
357C8—62no—
358C8—64no—
359C8—121no—
360C2×C3—22no—
361C2×C3#239211yescharacteristic
362C2×C3#306912yes—
363D3—81no—
364C22—22no—
365C22—122no—
366C22—121no—
367C22D3×#12111yescharacteristic
368C22#564912yes—
369C4—42no—
370C4—44no—
371C4—62no—
372C4—62no—
373C4—121no—
374C4—242no—
375C4#558611yescharacteristic
376C4#560612yes—
377C4#566811yescharacteristic
378C3#2372011yescharacteristic, complemented, Sylow
379C2—22no—
380C2—241no—
381C2#7689111yescharacteristic
382C2#8094412yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2047C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#2047D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2047

Complements

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

NH ≅ G/Ntype
1. C2×#5184C2semidirect
4. C3×#2047C2semidirect
6. #70533C2semidirect
7. #70337C2semidirect
9. Dic3×#101C2semidirect
11. #68917C2semidirect
51. #2047D3semidirect
378. C3#23720semidirect

Maximal subgroups

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

subgroupindex
C2×#51842
#790712
#789462
C3×#20472
#720742
#705332
#703372
#701262
Dic3×#1012
#689222
#689172
#237203