Finite Group Databasebeta

← #10690967

Subgroups — #10690967

Order 768.

692 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×C3×#382C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal
5#67612C212yesmaximal, complemented
6#9156—31nomaximal, Sylow
7C22×C3×C4⋊C4—21no—
8C22×C3×C42C2211yescharacteristic
9C2×#930C2211yescharacteristic
10C2×#929—21no—
11C3×#382—24no—
12C2×C3×#95—21no—
13C2×C3×#95C2212yes—
14#5213—21no—
15#5212C2211yescharacteristic
16C2×#830C2212yes—
17#4940—22no—
18#4939C412yescomplemented
19#4928C412yescomplemented
20#4524—22no—
21#4451—24no—
22#2854—31no—
23C2×#382D311yescharacteristic, complemented
24#1817—31no—
25C2×#327—32no—
26#1294—32no—
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—21no—
32C22×D12C2×C411yescharacteristic
33C2×C3×C4⋊C4—24no—
34C2×C3×C4⋊C4—42no—
35C2×C3×C4⋊C4—42no—
36C2×C3×C42—22no—
37C2×C3×C42—22no—
38C2×C3×C42C2×C412yes—
39C2×C3×C42D412yes—
40C2×#217—21no—
41#930—22no—
42#930—22no—
43#929—42no—
44#929—42no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—21no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×C4⋊Dic3C2×C412yes—
50C2×Dic3⋊C4—22no—
51C22×C3⋊C8—22no—
52C3×#95—24no—
53C3×#95—22no—
54C3×#95D414yes—
55C3⋊#95—22no—
56C3⋊#95—21no—
57#830—22no—
58#830—22no—
59C2×#123—61no—
60C2×#121—31no—
61C22×C4⋊C4—21no—
62C22×C42C2×D311yescharacteristic
63C2×C22⋊C8—32no—
64#390—32no—
65#389—32no—
66#388—62no—
67#386—61no—
68#382—24no—
69#380—31no—
70C2×#95—21no—
71C2×#95C2×D312yes—
72#336—62no—
73#327—64no—
74C24×C3C2×D411yescharacteristic
75C23×D3—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—24no—
83C22×C3×C4—42no—
84C22×C3×C4—42no—
85C22×C3×C4—44no—
86C22×C3×C4—44no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4D812yes—
92C22×C3×C4SD1612yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4#4112yes—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67—41no—
103C22×Dic3—21no—
104C22×Dic3—22no—
105C2×D12—22no—
106C2×D12—42no—
107C3×C4⋊C4—48no—
108C3×C42—24no—
109C3×C42—24no—
110#217—41no—
111#217—41no—
112(C2×D3)⋊C4—44no—
113(C2×D3)⋊C4—44no—
114(C2×D3)⋊C4—42no—
115C4⋊Dic3—22no—
116C4⋊Dic3—42no—
117Dic3⋊C4—44no—
118C2×C3⋊C8—44no—
119C2×C3⋊C8—42no—
120C22×D4—31no—
121C22×D4—61no—
122C23×C4—22no—
123C23×C4#6711yescharacteristic
124C23×C4#6711yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—62no—
127#123—122no—
128#123—122no—
129#121—62no—
130#121—62no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—62no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—61no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C4×D312yes—
144C2×C42D1212yes—
145C22⋊C8—62no—
146C22⋊C8—62no—
147#95—24no—
148#95—22no—
149#95—62no—
150#95—61no—
151#95#6714yes—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3C2×C22⋊C411yescharacteristic
155C23×C3#12011yescharacteristic
156C23×C3#12311yescharacteristic
157C23×C3C2×D811yescharacteristic
158C23×C3C2×SD1611yescharacteristic
159C23×C3AGL(1,ℤ/8)11yescharacteristic
160C23×C3#13711yescharacteristic
161C22×D3—41no—
162C22×D3—42no—
163C22×D3—41no—
164C22×D3—41no—
165C22×D3—42no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—84no—
185C2×C3×C4—84no—
186C2×C3×C4#10214yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189#67—81no—
190#67—81no—
191#67—81no—
192#67—81no—
193C2×Dic3—41no—
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199D12—82no—
200C3⋊C8—82no—
201C24—61no—
202C24C2×#6711yescharacteristic
203C2×D4—62no—
204C2×D4—122no—
205C2×D4—121no—
206C2×D4—121no—
207C2×D4—121no—
208C2×D4—121no—
209C2×D4—121no—
210C2×D4—121no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—62no—
225C22×C4—61no—
226C22×C4(C2×D3)⋊C412yes—
227C22×C4(C2×D3)⋊C412yes—
228C22×C4#21312yes—
229C22×C4#21512yes—
230C22×C4C2×C4×D311yescharacteristic
231C22×C4C2×D1211yescharacteristic
232C22×C4#23512yes—
233C22×C4C2×#6712yes—
234C2×C8—124no—
235C2×C8—122no—
236C4⋊C4—48no—
237C4⋊C4—62no—
238C4⋊C4—122no—
239C4⋊C4—124no—
240C22⋊C4—124no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C22⋊C4—124no—
244C22⋊C4—122no—
245C42—24no—
246C42—24no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3#38611yescharacteristic
256C22×C3#41012yes—
257C22×C3C2×#10211yescharacteristic
258C22×C3#41711yescharacteristic
259C22×C3#42112yes—
260C22×C3#44711yescharacteristic
261C22×C3#44811yescharacteristic
262C22×C3#48011yescharacteristic
263C22×C3#48111yescharacteristic
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—82no—
268C2×D3—82no—
269C3×C4—24no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—84no—
274Dic3—81no—
275Dic3—82no—
276C23—22no—
277C23—22no—
278C23—121no—
279C23—121no—
280C23—121no—
281C23—122no—
282C23—122no—
283C23C2×(C2×D3)⋊C411yescharacteristic
284C23#92911yescharacteristic
285C23C2×#21311yescharacteristic
286C23#93211yescharacteristic
287C23C2×#21511yescharacteristic
288C23#94211yescharacteristic
289C23#95311yescharacteristic
290D4—242no—
291D4—241no—
292D4—241no—
293D4—241no—
294D4—241no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4#80914yes—
322C2×C4C4×#6712yes—
323C2×C4#93012yes—
324C8—242no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#151212yes—
330C2×C3#151712yes—
331C2×C3#187411yescharacteristic
332C2×C3#189711yescharacteristic
333C2×C3#189911yescharacteristic
334D3—161no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—241no—
344C22—241no—
345C22—242no—
346C22—242no—
347C22—241no—
348C22C2×#80911yescharacteristic
349C22#496611yescharacteristic
350C22#496711yescharacteristic
351C22#496811yescharacteristic
352C22#500512yes—
353C22#500612yes—
354C22#521311yescharacteristic
355C22#521711yescharacteristic
356C22#522711yescharacteristic
357C4—24no—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—84no—
362C4—241no—
363C4—242no—
364C3#915611yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—481no—
370C2#6954412yes—
371C2#6954512yes—
372C2#7156011yescharacteristic
373C2#7159111yescharacteristic
374C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#382C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#382D3
hypocenterC3#9156
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
1. C2×C3×#382C2semidirect
3. #71531C2semidirect
5. #67612C2semidirect
18. #4939C4semidirect
19. #4928C4semidirect
23. C2×#382D3semidirect
364. C3#9156semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3822
#820802
#715312
C2×#44512
#676122
#91563