Finite Group Databasebeta

← #10712201

Subgroups — #10712201

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2912C211yesmaximal, characteristic, complemented
2#83805C211yesmaximal, characteristic
3#72006C211yesmaximal, characteristic, complemented
4#70656C211yesmaximal, characteristic, complemented
5C2×#4488C211yesmaximal, characteristic, complemented
6#67727C211yesmaximal, characteristic, complemented
7#67720C211yesmaximal, characteristic, complemented
8#13005—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10C3×#520—21no—
11#5822—21no—
12C2×#904C2211yescharacteristic
13C3×#428—21no—
14C3×#428—22no—
15C3×#427—21no—
16C3×#427C2211yescharacteristic
17C2×C3×#106—21no—
18C2×C3×#106C2211yescharacteristic
19#5143C412yescomplemented
20#5142C412yescomplemented
21#5141—22no—
22#5132—21no—
23C2×#819C2211yescharacteristic
24C2×#812C2211yescharacteristic
25C2×#812C2211yescharacteristic
26#4561—22no—
27#4488—21no—
28#4488—21no—
29#4488—22no—
30#3067—31no—
31#2912D311yescharacteristic, complemented
32#2145—31no—
33C2×#360—31no—
34#2117—31no—
35#1389—31no—
36#1383—31no—
37C2×C3×#41D411yescharacteristic
38C2×#235—21no—
39C3×#131—21no—
40C3×#131—21no—
41C3×#131C2×C412yes—
42C3×#131D412yes—
43C2×C3×M16—21no—
44C2×C3×M16D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8C2311yescharacteristic
47C3×C4×Q8—41no—
48C3×C4×D4—42no—
49C3×C4×D4—41no—
50C3×#117—22no—
51C3×#117—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—21no—
54#951—22no—
55#950—42no—
56#949—22no—
57#942—41no—
58#932—41no—
59C2×#208—21no—
60C2×Dic12C2×C411yescharacteristic, complemented
61#904—21no—
62#904—21no—
63#904—21no—
64#904—21no—
65C2×D24C2×C411yescharacteristic, complemented
66C2×#204—21no—
67C3×#107—22no—
68C3×#107—21no—
69C3×#106—22no—
70C3×#106—21no—
71C3×#106D411yescharacteristic
72C3×#106D412yes—
73C3×#106D411yescharacteristic
74#824—22no—
75#823—22no—
76#819—21no—
77#819C2×C412yescomplemented
78#812—21no—
79#812—21no—
80#812—21no—
81#812—21no—
82C2×C3⋊C16—21no—
83C2×C3⋊C16—21no—
84#575—61no—
85C2×#135—31no—
86C2×#131C2×D311yescharacteristic
87#520—21no—
88C2×M32—31no—
89C2×M32—31no—
90#473—32no—
91#472—32no—
92#471—62no—
93C2×#108—31no—
94#428—21no—
95#428—22no—
96#427—21no—
97#427C2×D311yescharacteristic
98C2×#106—21no—
99C2×#106C2×D311yescharacteristic
100#413—61no—
101#360—62no—
102#360—61no—
103#360—61no—
104#350—62no—
105C3×#41—21no—
106C3×#41—21no—
107C3×#41C22⋊C412yes—
108C3×#41SD1612yes—
109C2×C3×Q8D811yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4D811yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—42no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117#235—41no—
118#235—41no—
119#235—41no—
120#235—41no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C3×M16—21no—
125C3×M16—22no—
126C3×M16—22no—
127C3×M16C22⋊C412yes—
128C2×C3×C8—21no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8C22×C411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8#4111yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—41no—
144C3×C42—42no—
145C3×C42—41no—
146#216—42no—
147#215—42no—
148#214—42no—
149#213—42no—
150#208—22no—
151#208—41no—
152C2×C3⋊C8—41no—
153Dic12—21no—
154Dic12—21no—
155D24—21no—
156D24—21no—
157#204—41no—
158#204—41no—
159C3⋊C16—41no—
160C3⋊C16—41no—
161C2×#41—61no—
162C2×#41#6711yescharacteristic
163#137—62no—
164#137—121no—
165AGL(1,ℤ/8)—62no—
166AGL(1,ℤ/8)—121no—
167#135—61no—
168#135—61no—
169#135—61no—
170#135—61no—
171#135—122no—
172C2×Q16—31no—
173C2×SD16—61no—
174C2×D8—31no—
175#131—21no—
176#131—21no—
177#131C4×D312yes—
178#131D1212yes—
179C2×M16—21no—
180C2×M16—61no—
181C2×M16#6711yescharacteristic
182C22×C8—21no—
183C22×C8C22×D311yescharacteristic
184C4×Q8—41no—
185C4×D4—42no—
186C4×D4—41no—
187#117—22no—
188#117—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191M32—61no—
192M32—61no—
193M32—61no—
194M32—61no—
195C2×C16—61no—
196C2×C16—61no—
197#108—32no—
198#108—61no—
199#107—22no—
200#107—21no—
201#106—22no—
202#106—21no—
203#106#6711yescharacteristic
204#106#6712yes—
205#106#6711yescharacteristic
206#101—62no—
207#100—62no—
208C23×C3—21no—
209C23×C3AGL(1,ℤ/8)11yescharacteristic
210C22×D3—41no—
211C3×Q8—21no—
212C3×Q8#10212yes—
213C3×D4—22no—
214C3×D4—22no—
215C3×D4—21no—
216C3×D4#10212yes—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—161no—
224C2×C3×C4—162no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13711yescharacteristic
231#67—81no—
232#67—81no—
233C2×Dic3—41no—
234D12—41no—
235D12—41no—
236C4×D3—81no—
237C4×D3—81no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—22no—
241C3×C8C4×D411yescharacteristic
242C3×C8C4×D411yescharacteristic
243C3×C8#12111yescharacteristic
244C3×C8#12111yescharacteristic
245C3⋊C8—42no—
246#41—21no—
247#41—21no—
248#41—121no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41(C2×D3)⋊C412yes—
253#41#21512yes—
254C2×Q8—61no—
255C2×Q8#21311yescharacteristic
256C2×D4—21no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4#21311yescharacteristic
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—121no—
264C22×C4C2×#6711yescharacteristic
265Q16—61no—
266Q16—61no—
267Q16—122no—
268SD16—122no—
269SD16—122no—
270SD16—121no—
271SD16—121no—
272D8—61no—
273D8—61no—
274D8—122no—
275M16—21no—
276M16—22no—
277M16—22no—
278M16—62no—
279M16—121no—
280M16(C2×D3)⋊C412yes—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—121no—
285C2×C8C2×C4×D311yescharacteristic
286C2×C8C2×D1211yescharacteristic
287C2×C8#23511yescharacteristic
288C2×C8#23511yescharacteristic
289C2×C8C2×#6711yescharacteristic
290C2×C8C2×#6711yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C42—42no—
299C42—41no—
300C16—121no—
301C16—121no—
302C22×C3—22no—
303C22×C3—41no—
304C22×C3—42no—
305C22×C3#41711yescharacteristic
306C22×C3#44711yescharacteristic
307C22×C3#48111yescharacteristic
308C2×D3—81no—
309C2×D3—161no—
310C3×C4—22no—
311C3×C4—81no—
312C3×C4—82no—
313C3×C4—81no—
314C3×C4#38611yescharacteristic
315C3×C4C2×#10211yescharacteristic
316C3×C4#44811yescharacteristic
317C3×C4#48011yescharacteristic
318Dic3—81no—
319C23—21no—
320C23—121no—
321C23#93211yescharacteristic
322Q8—21no—
323Q8—121no—
324Q8—121no—
325Q8#80912yes—
326D4—22no—
327D4—22no—
328D4—21no—
329D4—121no—
330D4—121no—
331D4—241no—
332D4—241no—
333D4#80912yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—121no—
341C2×C4—161no—
342C2×C4—162no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4C2×(C2×D3)⋊C411yescharacteristic
346C2×C4#92911yescharacteristic
347C2×C4C2×#21311yescharacteristic
348C2×C4C2×#21511yescharacteristic
349C2×C4#94211yescharacteristic
350C2×C4#95311yescharacteristic
351C8—22no—
352C8—122no—
353C8C4×#6711yescharacteristic
354C8C4×#6711yescharacteristic
355C8#93011yescharacteristic
356C8#93011yescharacteristic
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—42no—
361C2×C3#187411yescharacteristic
362D3—161no—
363C22—22no—
364C22—41no—
365C22—42no—
366C22—241no—
367C22—481no—
368C22#496611yescharacteristic
369C22#496811yescharacteristic
370C22#521711yescharacteristic
371C4—22no—
372C4—82no—
373C4—81no—
374C4—81no—
375C4—241no—
376C4C2×#80911yescharacteristic
377C4#496711yescharacteristic
378C4#521311yescharacteristic
379C4#522711yescharacteristic
380C3#1300511yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—481no—
386C2#7156011yescharacteristic
subgroupG/H
centerC2#71560
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2912C2
socleC2×C3#1874
prime radical#83805C2
cosocleC22×C3×C8C23
hypercenter#2912D3
hypocenterC3#13005
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. C3×#2912C2semidirect
3. #72006C2semidirect
4. #70656C2semidirect
5. C2×#4488C2semidirect
6. #67727C2semidirect
7. #67720C2semidirect
19. #5143C4semidirect
20. #5142C4semidirect
31. #2912D3semidirect
60. C2×Dic12C2×C4semidirect
65. C2×D24C2×C4semidirect
77. #819C2×C4semidirect
380. C3#13005semidirect

Maximal subgroups

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

subgroupindex
C3×#29122
#838052
#720062
#706562
C2×#44882
#677272
#677202
#130053