Finite Group Databasebeta

← #10691115

Subgroups — #10691115

Order 768.

446 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2356C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3#71621C211yesmaximal, characteristic
4#69546C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#9178—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#391—21no—
12C3×#387—22no—
13C3×#383—21no—
14C3×#382C2211yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C3×#376—21no—
18#5225—21no—
19#5022—22no—
20#5021C412yescomplemented
21#5020C412yescomplemented
22#4964—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4535—22no—
27#4452—21no—
28#4452—21no—
29#4452—22no—
30#3078—31no—
31#2356D311yescharacteristic
32#1518—31no—
33#1503—31no—
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×Q8D411yescharacteristic
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8C2×C412yes—
41C3×C4×Q8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42—22no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51#924—21no—
52#924—21no—
53C2×#208—21no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×#95—22no—
60C3×#95—22no—
61C3×#95—22no—
62C3×#95—21no—
63C3×#95D411yescharacteristic
64C3×#95D411yescharacteristic
65C3×#95D412yes—
66#835—22no—
67#835—22no—
68#830—21no—
69#830—21no—
70#808—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)C2×C412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#479—62no—
81#477—32no—
82#475—32no—
83#431—31no—
84#431—31no—
85#428—61no—
86#423—31no—
87#419—61no—
88#391—21no—
89#387—22no—
90#383—21no—
91#382C2×D311yescharacteristic
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×D311yescharacteristic
94#376—21no—
95#328—61no—
96#328—61no—
97#328—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×Dic6—21no—
114C2×Dic6—21no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C42—22no—
123C3×C42—22no—
124C3×C42—44no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—41no—
140#208—41no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—41no—
144C22×Q8#6711yescharacteristic
145C2×M16—61no—
146C4⋊Q8—31no—
147C4⋊Q8—31no—
148#125—61no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8C4×D312yes—
156C4×Q8D1212yes—
157#117—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4#6711yescharacteristic
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#106—62no—
168C4⋊C8—32no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—62no—
173#103—62no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176C4×C8—61no—
177C4×C8—61no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—21no—
182#95#6711yescharacteristic
183#95#6711yescharacteristic
184#95#6712yes—
185C23×C3#12011yescharacteristic
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—41no—
189C3×Q8—41no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—41no—
218Dic6—81no—
219Dic6—81no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222C3⋊C8—81no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8(C2×D3)⋊C412yes—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4C2×#6711yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—121no—
240M16—121no—
241C2×C8—62no—
242C2×C8—121no—
243C2×C8—121no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4(C2×D3)⋊C412yes—
257C22⋊C4—121no—
258C42—22no—
259C42—22no—
260C42—44no—
261C42—41no—
262C42—41no—
263C42—121no—
264C42C2×C4×D311yescharacteristic
265C42C2×D1211yescharacteristic
266C42#23511yescharacteristic
267C42#23511yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#45211yescharacteristic
272C22×C3#45211yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—42no—
280C3×C4—81no—
281C3×C4—82no—
282C3×C4#41611yescharacteristic
283C3×C4#41911yescharacteristic
284C3×C4#48511yescharacteristic
285C3×C4#48511yescharacteristic
286Dic3—81no—
287Dic3—81no—
288C23#95311yescharacteristic
289Q8—22no—
290Q8—22no—
291Q8—41no—
292Q8—41no—
293Q8—241no—
294Q8—241no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4C2×(C2×D3)⋊C411yescharacteristic
314C2×C4C4×#6711yescharacteristic
315C2×C4C4×#6711yescharacteristic
316C2×C4#92911yescharacteristic
317C2×C4#93011yescharacteristic
318C2×C4#93011yescharacteristic
319C2×C4#95011yescharacteristic
320C2×C4#95011yescharacteristic
321C2×C4#95111yescharacteristic
322C2×C4#95111yescharacteristic
323C8—122no—
324C8—241no—
325C8—241no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187311yescharacteristic
329C2×C3#190511yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#521311yescharacteristic
334C22#523911yescharacteristic
335C22#523911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—81no—
344C4—82no—
345C4—241no—
346C4—241no—
347C4#499611yescharacteristic
348C4#499611yescharacteristic
349C4#500211yescharacteristic
350C4#500411yescharacteristic
351C3#917811yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#7158511yescharacteristic
355C2#7164511yescharacteristic
356C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2356C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2356D3
hypocenterC3#9178
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
20. #5021C4semidirect
21. #5020C4semidirect
351. C3#9178semidirect

Maximal subgroups

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

subgroupindex
C3×#23562
#824312
#716212
#695462
#694712
#676312
#676202
#91783