Finite Group Databasebeta

← #10691891

Subgroups — #10691891

Order 768.

470 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83227C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic
5#70515C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#9296—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—21no—
15#5094—22no—
16#5093C412yescomplemented
17#5080C412yescomplemented
18#4980—22no—
19#4980—22no—
20#4971—22no—
21#4936C2211yescharacteristic
22#4932C2211yescharacteristic
23#4925C2211yescharacteristic
24C3×#332—22no—
25#4555—21no—
26#4555—21no—
27#4467—22no—
28#3229—31no—
29#2327—31no—
30#1995—31no—
31#1546D311yescharacteristic
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C42C2311yescharacteristic
44Q8×Dic3—42no—
45Q8×Dic3—42no—
46#926—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3D411yescharacteristic
53C2×C4×Dic3D411yescharacteristic
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3⋊#95—24no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95D412yes—
69#830—21no—
70#810—24no—
71#808—24no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8C2×C412yes—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76C2×C4×Q8—61no—
77#498—32no—
78#487—32no—
79#483—62no—
80#475—62no—
81#431—31no—
82#431C2×D311yescharacteristic
83#423C2×D311yescharacteristic
84#391—61no—
85#387—62no—
86#387—62no—
87#382—31no—
88#376—31no—
89#339—61no—
90#339—61no—
91#332—22no—
92#328—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—22no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—22no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—22no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4C22⋊C414yes—
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120#217—41no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—44no—
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—44no—
134C4×Dic3—41no—
135C4×Dic3SD1612yes—
136C2×C3⋊C8—22no—
137C22×Q8—21no—
138C2×M16—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143C4⋊Q8C4×D312yes—
144C4⋊Q8#6712yes—
145#125—61no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—122no—
151C4×Q8—122no—
152#117—61no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4D1212yes—
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—62no—
160C2×C42C22×D311yescharacteristic
161#106—64no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—61no—
166#103—64no—
167C22⋊C8—21no—
168C22⋊C8—61no—
169C4×C8—21no—
170C4×C8—32no—
171#95—32no—
172#95—64no—
173#95—61no—
174#95—61no—
175#95—61no—
176C23×C3#12011yescharacteristic
177C3×Q8—42no—
178C3×Q8—42no—
179C3×Q8—42no—
180C3×Q8—42no—
181C2×C3×C4—24no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207Dic6—42no—
208Dic6—81no—
209C3×C8—81no—
210C3×C8—81no—
211C3⋊C8—42no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—61no—
219C2×Q8—61no—
220C22×C4—22no—
221C22×C4—61no—
222C22×C4—62no—
223C22×C4—61no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227M16—41no—
228M16—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—62no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—124no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4(C2×D3)⋊C414yes—
245C22⋊C4—121no—
246C42—32no—
247C42—61no—
248C42—61no—
249C42—61no—
250C42—124no—
251C42—121no—
252C42C2×C4×D311yescharacteristic
253C42#23711yescharacteristic
254C42#23711yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C22×C3#45011yescharacteristic
260C22×C3#45211yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—44no—
266C3×C4C4×SD1611yescharacteristic
267C3×C4#44011yescharacteristic
268C3×C4#46111yescharacteristic
269C3×C4#46311yescharacteristic
270Dic3—42no—
271Dic3—82no—
272Dic3—81no—
273Dic3—81no—
274Dic3—81no—
275C23#88211yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—42no—
279Q8—42no—
280Q8—122no—
281Q8—241no—
282C2×C4—24no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—62no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—122no—
291C2×C4—121no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—122no—
297C2×C4—121no—
298C2×C4#67⋊C411yescharacteristic
299C2×C4#67⋊C411yescharacteristic
300C2×C4#88611yescharacteristic
301C2×C4#91111yescharacteristic
302C2×C4#91211yescharacteristic
303C2×C4D3×SD1611yescharacteristic
304C2×C4#91511yescharacteristic
305C2×C4C2×(C2×D3)⋊C411yescharacteristic
306C2×C4#93911yescharacteristic
307C2×C4#93911yescharacteristic
308C8—81no—
309C8—81no—
310C8—122no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#187711yescharacteristic
314C2×C3#190511yescharacteristic
315C2×C3#191211yescharacteristic
316C22—21no—
317C22—21no—
318C22#477611yescharacteristic
319C22#478311yescharacteristic
320C22#495811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—122no—
327C4—242no—
328C4—241no—
329C4—241no—
330C4—241no—
331C4#475711yescharacteristic
332C4#475811yescharacteristic
333C4#515311yescharacteristic
334C4#516211yescharacteristic
335C3#929611yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2#7010511yescharacteristic
339C2#7017111yescharacteristic
340C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9296
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
16. #5093C4semidirect
17. #5080C4semidirect
335. C3#9296semidirect

Maximal subgroups

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

subgroupindex
#832272
#832232
C3×#15462
#707932
#705152
#677372
#677132
#92963