Finite Group Databasebeta

← #10693196

Subgroups — #10693196

Order 768.

520 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#2973C211yesmaximal, characteristic
2#82387C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#69971C212yesmaximal, complemented
5#67685C212yesmaximal, complemented
6#9517—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×Dic3⋊C4C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407C2212yes—
12C2×C3×C22⋊C8—21no—
13#5220C411yescharacteristic
14#5220C411yescharacteristic
15#5212C2211yescharacteristic
16#5210—21no—
17#5207C2212yes—
18#4982—21no—
19#4948—21no—
20#4946C411yescharacteristic
21#4946C411yescharacteristic
22#4933—21no—
23#4528—21no—
24#4525—21no—
25#4469—24no—
26#2973D311yescharacteristic
27#2579—31no—
28#1465—31no—
29#1463—32no—
30#1305—32no—
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×Dic3C2×C411yescharacteristic
35C23×Dic3C2×C411yescharacteristic
36C2×C3×M16—22no—
37C22×C3×C8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217C2×C411yescharacteristic
47C2×#217C2×C411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—22no—
53C2×Dic3⋊C4—22no—
54C2×Dic3⋊C4—21no—
55C2×#208—22no—
56C3×C4⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8D414yes—
60C3×#97—22no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—22no—
67#830—22no—
68C2×C4×D4C2×D311yescharacteristic
69C22×C4⋊C4—31no—
70#435—24no—
71#432—21no—
72#407—32no—
73#407C2×D312yes—
74C2×C22⋊C8—21no—
75#398—61no—
76#396—61no—
77#387—61no—
78#386—31no—
79#386—31no—
80#385—31no—
81#385—31no—
82#385—61no—
83#380—31no—
84#324—61no—
85#324—64no—
86#323—61no—
87C24×C3C22⋊C411yescharacteristic
88C24×C3C22×C411yescharacteristic
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—81no—
95C22×C3×C4—81no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C22×Dic3—21no—
104C22×Dic3—22no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—22no—
109C22×Dic3—42no—
110C22×Dic3—81no—
111C3×M16—82no—
112C2×C3×C8—24no—
113C2×C3×C8—82no—
114C3×C4⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C42—21no—
118C3×C42—21no—
119#217—42no—
120#217—41no—
121#217—41no—
122#217—81no—
123C4⋊Dic3—81no—
124Dic3⋊C4—42no—
125Dic3⋊C4—41no—
126Dic3⋊C4—42no—
127Dic3⋊C4—41no—
128Dic3⋊C4—81no—
129#208—82no—
130C2×C3⋊C8—42no—
131C22×D4C4×D311yescharacteristic
132C23×C4—31no—
133C23×C4—31no—
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C2×M16—22no—
137C2×M16—62no—
138C22×C8—22no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4D1211yescharacteristic
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4C4×D311yescharacteristic
154C2×C22⋊C4D1211yescharacteristic
155C2×C42#6711yescharacteristic
156C4⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—62no—
160C22⋊C8—62no—
161C22⋊C8#6714yes—
162#97—22no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3#10112yes—
177C23×C3C2×C22⋊C411yescharacteristic
178C23×C3C4×D412yes—
179C3×D4—44no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—24no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—82no—
211C3×C8—44no—
212C3⋊C8—82no—
213C24(C2×D3)⋊C411yescharacteristic
214C24C2×C4×D311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—61no—
221C22×C4—62no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—122no—
229C22×C4—241no—
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4C2×D1211yescharacteristic
234C22×C4#23512yes—
235C22×C4C2×#6711yescharacteristic
236C22×C4C2×#6712yes—
237M16—82no—
238M16—242no—
239C2×C8—24no—
240C2×C8—82no—
241C2×C8—122no—
242C4⋊C4—22no—
243C4⋊C4—122no—
244C4⋊C4—121no—
245C4⋊C4—122no—
246C4⋊C4—121no—
247C4⋊C4—241no—
248C4⋊C4—241no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—122no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—241no—
255C42—21no—
256C42—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—81no—
269C22×C3#38611yescharacteristic
270C22×C3C2×#10111yescharacteristic
271C22×C3#41311yescharacteristic
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278Dic3—42no—
279Dic3—81no—
280Dic3—81no—
281Dic3—81no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#82412yes—
291C23C2×(C2×D3)⋊C411yescharacteristic
292C23C4×#6712yes—
293D4—44no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—64no—
308C2×C4—81no—
309C2×C4—121no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—242no—
322C2×C4#92911yescharacteristic
323C2×C4#93012yes—
324C2×C4#95311yescharacteristic
325C8—44no—
326C8—242no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3#189111yescharacteristic
332C2×C3#190111yescharacteristic
333C2×C3#191311yescharacteristic
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22#513211yescharacteristic
347C22C2×#82411yescharacteristic
348C22#521311yescharacteristic
349C4—21no—
350C4—22no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—122no—
356C4—241no—
357C4—241no—
358C4—241no—
359C3#951711yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#7165611yescharacteristic
365C2#7200311yescharacteristic
366C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9517
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #69971C2semidirect
5. #67685C2semidirect
359. C3#9517semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823872
#716022
#699712
#676852
#95173