Finite Group Databasebeta

← #10824912

Subgroups — #10824912

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
1#79200C212yesmaximal
2C2×#4853C211yesmaximal, characteristic, complemented
3#79071C211yesmaximal, characteristic
4C3×#2090C211yesmaximal, characteristic, complemented
5#70679C212yesmaximal, complemented
6#70593C211yesmaximal, characteristic
7#70543C211yesmaximal, characteristic, complemented
8#68987C212yesmaximal
9#68973C212yesmaximal, complemented
10#68944C211yesmaximal, characteristic
11#68917C211yesmaximal, characteristic, complemented
12#23720—31nomaximal, Sylow
13C2×#909C2212yes—
14C22×Dic12C2211yescharacteristic
15#5522C2211yescharacteristic
16C2×#887C2212yes—
17C3×#428C2212yes—
18C2×C3×#106C2211yescharacteristic
19C3×#424C2211yescharacteristic
20C2×#824C2212yes—
21C2×#824C2212yes—
22#5129C2211yescharacteristic
23C2×#822C2212yes—
24Dic3×M16C2212yes—
25Dic3×M16C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×#815C2211yescharacteristic
28#5004C2212yes—
29#4999C2211yescharacteristic
30C2×#810C2212yes—
31#4966C2212yes—
32#4889—22no—
33#4873—22no—
34#4853—22no—
35#4853—22no—
36#4853—22no—
37#4852—22no—
38#4840—22no—
39#4835—22no—
40C3×#343C2212yes—
41C3×#341C2211yescharacteristic
42#4552C2211yescharacteristic
43#4530C2212yes—
44C4×#137—32no—
45C2×#441—31no—
46#2923—31no—
47#2090D311yescharacteristic, complemented
48#2079—31no—
49#2064—32no—
50#2052—32no—
51#2047—31no—
52#2020—31no—
53#1778—31no—
54C4×#101—32no—
55C2×#235C2311yescharacteristic
56C22×Dic6C2312yes—
57C2×C3×M16C2311yescharacteristic
58C2×C3×M16C2312yes—
59C22×C3×C8C2311yescharacteristic
60C3×#117C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—22no—
64C2×C4×Dic3C2311yescharacteristic
65C2×#208C2312yes—
66C2×#208C2311yescharacteristic
67C22×C3⋊C8C2311yescharacteristic
68#909—24no—
69#909—24no—
70C2×Dic12—22no—
71C2×Dic12—24no—
72C2×Dic12—22no—
73C2×#204—22no—
74#893D412yes—
75#887—22no—
76#887—22no—
77#887—22no—
78#887—22no—
79#887D412yes—
80#67⋊C4—22no—
81#872—21no—
82C42×D3—21no—
83C3×#107—22no—
84C3×#106—22no—
85C3×#106—22no—
86C3×C4⋊C8—22no—
87#824—24no—
88#824—24no—
89#822—24no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#815—22no—
93#815—21no—
94#815D412yes—
95#815D412yes—
96C8×Dic3—22no—
97C8×Dic3—21no—
98#810—22no—
99#810—22no—
100#810—22no—
101#810—24no—
102#809—22no—
103C4⋊(C3⋊C8)—22no—
104C2×#137—32no—
105C22×Q16—31no—
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#428C2×D312yes—
118C2×#106C2×D311yescharacteristic
119#424—31no—
120#424C2×D311yescharacteristic
121#419—32no—
122#417—32no—
123C2×#103—32no—
124C2×#101—32no—
125C2×#101—32no—
126#408—31no—
127#408—31no—
128C4×M16—31no—
129C4×M16—32no—
130C2×#97—31no—
131#343—32no—
132#343C2×D312yes—
133#341—31no—
134#341C2×D311yescharacteristic
135C22×C3×C4—22no—
136C22×C3×C4C2411yescharacteristic
137C2×#67—21no—
138C22×Dic3—22no—
139C22×Dic3#4112yes—
140#235—22no—
141#235—22no—
142C2×D12C2×D411yescharacteristic
143C2×C4×D3—21no—
144C2×C4×D3—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—22no—
150C2×Dic6C2×D411yescharacteristic
151C2×Dic6#4114yes—
152C3×M16—22no—
153C3×M16—22no—
154C3×M16#4114yes—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—21no—
158C2×C3×C8C2×D412yes—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4C2×D412yes—
162C3×C22⋊C4—21no—
163C3×C42—21no—
164(C2×D3)⋊C4—22no—
165Dic3⋊C4—22no—
166Dic3⋊C4—42no—
167Dic3⋊C4—42no—
168C4×Dic3—22no—
169C4×Dic3—42no—
170C4×Dic3—42no—
171C4×Dic3C2×D412yes—
172C4×Dic3C2×D411yescharacteristic
173C4×Dic3C2×D411yescharacteristic
174#208—22no—
175#208—22no—
176#208—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8C2×D412yes—
181Dic12—28no—
182Dic12—28no—
183#204—28no—
184C2×#41—31no—
185C22×Q8—32no—
186#137—64no—
187#137—64no—
188C2×Q16—64no—
189C2×Q16—62no—
190C2×Q16—62no—
191C2×SD16—62no—
192C2×M16—32no—
193C2×M16—31no—
194C2×M16C22×D311yescharacteristic
195C2×M16C22×D312yes—
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—62no—
208C2×C4⋊C4C22×D312yes—
209C2×C42—31no—
210C2×C42—61no—
211C2×C42—62no—
212#108—64no—
213#107—22no—
214#106—22no—
215#106—22no—
216C4⋊C8—22no—
217C4⋊C8—62no—
218#103—64no—
219#103—62no—
220#103—62no—
221#103—62no—
222#102—62no—
223#101—64no—
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—
235C2×C3×C4—22no—
236C2×C3×C4—42no—
237C2×C3×C4—82no—
238C2×C3×C4C22×D411yescharacteristic
239C2×C3×C4C22×D411yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4C2×#4112yes—
242#67—42no—
243C2×Dic3—21no—
244C2×Dic3—24no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3#12112yes—
249C2×Dic3#12112yes—
250D12—22no—
251C4×D3—41no—
252C4×D3—41no—
253C4×D3—82no—
254Dic6—24no—
255Dic6—24no—
256Dic6—22no—
257Dic6—42no—
258Dic6—42no—
259C3×C8—24no—
260C3×C8—22no—
261C3×C8—41no—
262C3⋊C8—22no—
263C3⋊C8—42no—
264C3⋊C8—41no—
265#41—62no—
266#41—62no—
267C2×Q8—31no—
268C2×Q8—34no—
269C2×Q8—62no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×D4—31no—
275C2×D4—61no—
276C22×C4—22no—
277C22×C4—32no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4C23×D311yescharacteristic
282Q16—68no—
283Q16—68no—
284SD16—68no—
285M16—22no—
286M16—22no—
287M16—62no—
288M16—62no—
289M16—62no—
290M16#23914yes—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—32no—
295C2×C8—62no—
296C2×C8—61no—
297C2×C8—62no—
298C2×C8D3×D412yes—
299C4⋊C4—22no—
300C4⋊C4—22no—
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—42no—
320C3×C4—82no—
321C3×C4#52511yescharacteristic
322C3×C4#52512yes—
323C3×C4#53211yescharacteristic
324Dic3—22no—
325Dic3—22no—
326Dic3—44no—
327Dic3—41no—
328C23—61no—
329C23D3×#4111yescharacteristic
330Q8—64no—
331Q8—64no—
332Q8—62no—
333Q8—122no—
334Q8—122no—
335D4—62no—
336D4—122no—
337C2×C4—22no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—42no—
341C2×C4—64no—
342C2×C4—61no—
343C2×C4—82no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—242no—
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×D3×D411yescharacteristic
352C2×C4C2×#23912yes—
353C2×C4D3×#4112yes—
354C8—22no—
355C8—24no—
356C8—41no—
357C8—62no—
358C8—121no—
359C8—122no—
360C2×C3—22no—
361C2×C3#239211yescharacteristic
362C2×C3#306912yes—
363D3—81no—
364C22—22no—
365C22—122no—
366C22—121no—
367C22D3×#12111yescharacteristic
368C22#569112yes—
369C4—42no—
370C4—62no—
371C4—62no—
372C4—82no—
373C4—121no—
374C4—124no—
375C4#562712yes—
376C4#565611yescharacteristic
377C4#572511yescharacteristic
378C3#2372011yescharacteristic, complemented, Sylow
379C2—22no—
380C2—241no—
381C2#7773711yescharacteristic
382C2#8093112yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2090C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#2090D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2090

Complements

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

NH ≅ G/Ntype
2. C2×#4853C2semidirect
4. C3×#2090C2semidirect
5. #70679C2semidirect
7. #70543C2semidirect
9. #68973C2semidirect
11. #68917C2semidirect
47. #2090D3semidirect
378. C3#23720semidirect

Maximal subgroups

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

subgroupindex
#792002
C2×#48532
#790712
C3×#20902
#706792
#705932
#705432
#689872
#689732
#689442
#689172
#237203