Finite Group Databasebeta

← #10691101

Subgroups — #10691101

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2351C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3#71621C211yesmaximal, characteristic
4#69546C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#9176—31nomaximal, Sylow
9C2×C3×C4×Q8—21no—
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#389—21no—
13C3×#389—22no—
14C3×#382—21no—
15C3×#382C2211yescharacteristic
16C3×C4×C4⋊C4—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
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#2351D311yescharacteristic
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×Q8—42no—
41C3×C4×Q8—42no—
42C3×C4×Q8C2×C412yes—
43C3×C4×Q8D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42—22no—
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58#870—21no—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×#95—22no—
64C3×#95—21no—
65C3×#95D411yescharacteristic
66C3×#95D411yescharacteristic
67C3×#95D412yes—
68#835—22no—
69#835—22no—
70#830—21no—
71#830—21no—
72#808—22no—
73#807—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2×D311yescharacteristic
83#479—62no—
84#477—32no—
85#475—32no—
86#431—31no—
87#431—31no—
88#428—61no—
89#423—31no—
90#419—61no—
91#389—21no—
92#389—22no—
93#382—21no—
94#382C2×D311yescharacteristic
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2×D311yescharacteristic
97#328—61no—
98#328—61no—
99#328—62no—
100Q8⋊C8—62no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8C22⋊C412yes—
105C2×C3×Q8SD1612yes—
106C2×C3×Q8Q1612yes—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×Dic6—21no—
116C2×Dic6—21no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—44no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4C22⋊C412yes—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42—44no—
129C3×C42—41no—
130C3×C42—41no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143#208—41no—
144#208—41no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C22×Q8#6711yescharacteristic
149C2×M16—61no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152#125—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—61no—
160C4×Q8—61no—
161C4×Q8C4×D312yes—
162C4×Q8D1212yes—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4#6711yescharacteristic
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#106—62no—
176C4⋊C8—32no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—62no—
181#103—62no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184C4×C8—61no—
185C4×C8—61no—
186#95—22no—
187#95—21no—
188#95#6711yescharacteristic
189#95#6711yescharacteristic
190#95#6712yes—
191C23×C3#12011yescharacteristic
192C3×Q8—22no—
193C3×Q8—22no—
194C3×Q8—22no—
195C3×Q8#10314yes—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×C3×C4C2×SD1611yescharacteristic
219C2×C3×C4C2×Q1611yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×Dic3—41no—
223C2×Dic3—41no—
224Dic6—81no—
225Dic6—81no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C3⋊C8—81no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8(C2×D3)⋊C412yes—
235C2×Q8#21412yes—
236C2×Q8#21612yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4C2×#6711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245M16—121no—
246M16—121no—
247C2×C8—62no—
248C2×C8—121no—
249C2×C8—121no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4(C2×D3)⋊C412yes—
265C22⋊C4—121no—
266C42—22no—
267C42—22no—
268C42—44no—
269C42—41no—
270C42—41no—
271C42—121no—
272C42C2×C4×D311yescharacteristic
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C42#23511yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#45011yescharacteristic
280C22×C3#45111yescharacteristic
281C3×C4—24no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—42no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4C2×#10311yescharacteristic
291C3×C4#41711yescharacteristic
292C3×C4#48311yescharacteristic
293C3×C4#48411yescharacteristic
294Dic3—81no—
295Dic3—81no—
296C23#95311yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—241no—
301Q8—241no—
302Q8#81014yes—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4C2×(C2×D3)⋊C411yescharacteristic
322C2×C4C4×#6711yescharacteristic
323C2×C4C4×#6711yescharacteristic
324C2×C4#92911yescharacteristic
325C2×C4#93011yescharacteristic
326C2×C4#93011yescharacteristic
327C2×C4#93211yescharacteristic
328C2×C4C2×#21411yescharacteristic
329C2×C4#94211yescharacteristic
330C2×C4C2×#21611yescharacteristic
331C8—122no—
332C8—241no—
333C8—241no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#187211yescharacteristic
337C2×C3#190511yescharacteristic
338C2×C3#190611yescharacteristic
339C22—21no—
340C22—21no—
341C22#521311yescharacteristic
342C22#521811yescharacteristic
343C22#522811yescharacteristic
344C4—24no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—42no—
351C4—82no—
352C4—81no—
353C4—241no—
354C4—241no—
355C4#496611yescharacteristic
356C4C2×#81011yescharacteristic
357C4#497011yescharacteristic
358C4#497111yescharacteristic
359C3#917611yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2#7155911yescharacteristic
363C2#7164511yescharacteristic
364C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2351C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2351D3
hypocenterC3#9176
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
20. #5021C4semidirect
21. #5020C4semidirect
359. C3#9176semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
#824312
#716212
#695462
#694712
#676312
#676202
#91763