Finite Group Databasebeta

← #10691844

Subgroups — #10691844

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2932C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#71814C211yesmaximal, characteristic
4#70508C211yesmaximal, characteristic, complemented
5#69828C211yesmaximal, characteristic, complemented
6#67713C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9289—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×#532C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#435—22no—
13C3×#435—22no—
14C3×#423C2211yescharacteristic
15C3×#408—21no—
16C2×C3×#97C2211yescharacteristic
17#5093C412yescomplemented
18#5076C412yescomplemented
19#5040—22no—
20#5019—22no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4551—21no—
25#4551—21no—
26#4537—22no—
27#4486—22no—
28#3206—31no—
29#2932D311yescharacteristic
30#1993—31no—
31#1549—31no—
32#1517—31no—
33#1286—31no—
34#1275—31no—
35C2×C3×#41—21no—
36C22×Dic6—21no—
37C3×#131—42no—
38C3×#131—42no—
39C2×C3×M16—22no—
40C2×C3×M16—21no—
41C22×C3×C8—22no—
42C22×C3×C8D411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121D412yes—
49C3×C4×D4—21no—
50C3×C4×D4D412yes—
51C2×C3×C42C2311yescharacteristic
52C2×#214—24no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869C2×C412yes—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D412yes—
62C3×C22⋊C8—24no—
63C3×#97—21no—
64C3×#97—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—21no—
68#830—21no—
69#810—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74C2×#131—21no—
75#532C2×D311yescharacteristic
76C2×C4⋊Q8—31no—
77#492—32no—
78#487—32no—
79#461—62no—
80#448—62no—
81#435—22no—
82#435—22no—
83#431—31no—
84#431—31no—
85#423C2×D311yescharacteristic
86#408—21no—
87C2×#97C2×D311yescharacteristic
88#382—31no—
89#340—61no—
90#340—61no—
91#332—62no—
92#327—62no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C2×C3×Q8—21no—
98C2×C3×D4—22no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4C22⋊C412yes—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—41no—
107C22×Dic3—41no—
108C2×Dic6—22no—
109C2×Dic6—41no—
110C2×Dic6—82no—
111C3×M16—44no—
112C3×M16—41no—
113C3×M16—41no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C2×C3×C8—22no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—44no—
121C2×C3×C8SD1612yescomplemented
122C3×C4⋊C4—21no—
123C3×C4⋊C4C22⋊C412yes—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#214—84no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×#41—21no—
140C22×Q8—61no—
141C2×SD16—64no—
142#131—42no—
143#131—42no—
144C2×M16—22no—
145C2×M16—21no—
146C22×C8—22no—
147C22×C8#6711yescharacteristic
148C22×C8#6711yescharacteristic
149C4⋊Q8—32no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8C4×D311yescharacteristic
154#127C4×D311yescharacteristic
155#124—21no—
156#121—21no—
157#121D1212yes—
158C4×D4—21no—
159C4×D4#6712yes—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C42C22×D311yescharacteristic
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8#6712yes—
168#103—64no—
169C22⋊C8—24no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172#97—21no—
173#97—21no—
174C4×C8—32no—
175C4×C8—61no—
176#95—61no—
177#95—61no—
178C23×C3—22no—
179C23×C3#12111yescharacteristic
180C3×Q8—41no—
181C3×Q8—41no—
182C3×D4—42no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—41no—
186C3×D4—41no—
187C3×D4—81no—
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4C2×C22⋊C411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×C3×C4C2×SD1611yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×Dic3—42no—
204C2×Dic3—82no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208Dic6—44no—
209Dic6—82no—
210C3×C8—22no—
211C3×C8—42no—
212C3×C8—41no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215#41—41no—
216#41—41no—
217#41—41no—
218#41—41no—
219C2×Q8—21no—
220C2×Q8—62no—
221C2×Q8—121no—
222C2×Q8—242no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4(C2×D3)⋊C412yes—
227C22×C4—22no—
228C22×C4—121no—
229C22×C4—121no—
230C22×C4C2×D1211yescharacteristic
231C22×C4#23511yescharacteristic
232C22×C4C2×#6711yescharacteristic
233SD16—244no—
234M16—44no—
235M16—41no—
236M16—41no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—44no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8#21412yescomplemented
247C4⋊C4—21no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4(C2×D3)⋊C412yes—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C42C2×C4×D311yescharacteristic
258C42#23511yescharacteristic
259C42C2×#6711yescharacteristic
260C42C2×#6711yescharacteristic
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3#38611yescharacteristic
267C22×C3#46111yescharacteristic
268C22×C3#46311yescharacteristic
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—42no—
274C3×C4C4×SD1611yescharacteristic
275C3×C4#44011yescharacteristic
276C3×C4#45011yescharacteristic
277C3×C4#45211yescharacteristic
278Dic3—82no—
279Dic3—161no—
280C23—22no—
281C23#93011yescharacteristic
282Q8—41no—
283Q8—41no—
284Q8—124no—
285Q8—242no—
286D4—42no—
287D4—42no—
288D4—42no—
289D4—41no—
290D4—41no—
291D4—81no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—122no—
298C2×C4—242no—
299C2×C4—241no—
300C2×C4—241no—
301C2×C4—241no—
302C2×C4C2×(C2×D3)⋊C411yescharacteristic
303C2×C4C4×#6711yescharacteristic
304C2×C4C4×#6711yescharacteristic
305C2×C4#92911yescharacteristic
306C2×C4#93011yescharacteristic
307C2×C4#93211yescharacteristic
308C2×C4C2×#21411yescharacteristic
309C2×C4#95011yescharacteristic
310C2×C4#95111yescharacteristic
311C2×C4#95311yescharacteristic
312C8—22no—
313C8—42no—
314C8—41no—
315C8—122no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—82no—
320C2×C3#187711yescharacteristic
321C2×C3#190211yescharacteristic
322C2×C3#191011yescharacteristic
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22#501611yescharacteristic
329C22#501911yescharacteristic
330C22#521311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—242no—
337C4—481no—
338C4C4×#21411yescharacteristic
339C4#501811yescharacteristic
340C4#521811yescharacteristic
341C4#523911yescharacteristic
342C3#928911yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—82no—
346C2#7161211yescharacteristic
347C2#7202211yescharacteristic
348C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2932C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#2932D3
hypocenterC3#9289
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
4. #70508C2semidirect
5. #69828C2semidirect
6. #67713C2semidirect
7. #67660C2semidirect
17. #5093C4semidirect
18. #5076C4semidirect
121. C2×C3×C8SD16semidirect
246. C2×C8#214semidirect
342. C3#9289semidirect

Maximal subgroups

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

subgroupindex
C3×#29322
#833172
#718142
#705082
#698282
#677132
#676602
#92893