Finite Group Databasebeta

← #10690977

Subgroups — #10690977

Order 768.

980 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 139 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×#381C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9158—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C2×#953—21no—
10C2×#930C412yescomplemented
11C2×#930C2211yescharacteristic
12C3×#381—24no—
13C2×C3×#95—21no—
14C2×C3×#95C2212yes—
15#5213—21no—
16#5213—22no—
17#5212C412yescomplemented
18#5212C2211yescharacteristic
19C2×#830—22no—
20C2×#830C2212yes—
21#4451—24no—
22#2847—31no—
23C2×#381D311yescharacteristic, complemented
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C412yes—
33C23×C3×C4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C22×#67—22no—
37C22×#67—21no—
38C22×D12C2×C411yescharacteristic
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—44no—
41#953—44no—
42C2×#217—22no—
43C2×#217—21no—
44#930—22no—
45#930—22no—
46#930—44no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C412yes—
51C22×C3⋊C8—22no—
52C3×#95—24no—
53C3×#95—22no—
54C3×#95D414yescomplemented
55C3⋊#95—22no—
56C3⋊#95—21no—
57#830—22no—
58#830—22no—
59#830—44no—
60C24×C4C2×D311yescharacteristic
61C2×#121—31no—
62C2×#121—32no—
63C2×#120—61no—
64C22×C22⋊C4—21no—
65C2×C22⋊C8—32no—
66C2×C22⋊C8—62no—
67#386—62no—
68#386—61no—
69#381—24no—
70#380—32no—
71#380—31no—
72C2×#95—21no—
73C2×#95C2×D312yes—
74#327—64no—
75C24×C3—24no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—22no—
80C24×C3—42no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C23×D3—21no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—48no—
96C22×C3×C4—48no—
97C22×C3×C4—44no—
98C22×C3×C4—44no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4C22⋊C412yes—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4#4112yes—
106C2×#67—44no—
107C2×#67—44no—
108C2×#67—44no—
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—42no—
112C2×#67—42no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C2×D12—22no—
116C2×D12—42no—
117C3×C22⋊C4—48no—
118#217—44no—
119#217—42no—
120(C2×D3)⋊C4—44no—
121(C2×D3)⋊C4—42no—
122C4⋊Dic3—22no—
123C4⋊Dic3—42no—
124C2×C3⋊C8—44no—
125C2×C3⋊C8—42no—
126C25#6711yescharacteristic
127C22×D4—31no—
128C22×D4—62no—
129C22×D4—61no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—44no—
134C23×C4C4×D312yes—
135C23×C4D1212yes—
136C23×C4#6711yescharacteristic
137C23×C4C22×D311yescharacteristic
138C22×C8—62no—
139#121—62no—
140#121—62no—
141#121—124no—
142#120—124no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—32no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4—61no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153C22⋊C8—124no—
154#95—24no—
155#95—22no—
156#95—62no—
157#95—61no—
158#95#6714yescomplemented
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—24no—
162C23×C3—24no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—48no—
170C23×C3—48no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3#9914yes—
176C23×C3C2×C22⋊C411yescharacteristic
177C23×C3#12011yescharacteristic
178C23×C3#12311yescharacteristic
179C23×C3#13512yes—
180C23×C3AGL(1,ℤ/8)12yes—
181C22×D3—41no—
182C22×D3—42no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—42no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—48no—
194C2×C3×C4—48no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—84no—
201C2×C3×C4—84no—
202C2×C3×C4C4×D412yes—
203C2×C3×C4#12112yes—
204#67—88no—
205#67—82no—
206#67—82no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213D12—82no—
214C3⋊C8—82no—
215C24—24no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—42no—
221C24—61no—
222C24(C2×D3)⋊C412yes—
223C24C2×#6711yescharacteristic
224C2×D4—62no—
225C2×D4—122no—
226C2×D4—124no—
227C2×D4—124no—
228C2×D4—124no—
229C2×D4—121no—
230C2×D4—122no—
231C2×D4—122no—
232C2×D4—121no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4(C2×D3)⋊C412yes—
253C22×C4C2×C4×D311yescharacteristic
254C22×C4C2×D1211yescharacteristic
255C22×C4#23512yes—
256C22×C4C2×#6712yes—
257C2×C8—124no—
258C2×C8—122no—
259C4⋊C4—62no—
260C4⋊C4—122no—
261C22⋊C4—48no—
262C22⋊C4—124no—
263C22⋊C4—124no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—48no—
276C22×C3—48no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3#38611yescharacteristic
282C22×C3C2×#9912yes—
283C22×C3#41611yescharacteristic
284C22×C3#41811yescharacteristic
285C22×C3#42112yes—
286C22×C3#44912yes—
287C22×C3#48212yes—
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—82no—
292C2×D3—82no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—84no—
297Dic3—81no—
298Dic3—82no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—122no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—122no—
320C23#80614yes—
321C23C2×(C2×D3)⋊C411yescharacteristic
322C23#92911yescharacteristic
323C23#94912yes—
324C23#95012yes—
325C23#95311yescharacteristic
326D4—242no—
327D4—248no—
328D4—242no—
329D4—242no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4C4×#6712yes—
353C2×C4#93012yes—
354C8—242no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—44no—
358C2×C3#151012yes—
359C2×C3#151612yes—
360C2×C3#187511yescharacteristic
361C2×C3#189711yescharacteristic
362C2×C3#189811yescharacteristic
363D3—161no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—24no—
368C22—21no—
369C22—21no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—48no—
374C22—48no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—241no—
380C22—242no—
381C22—242no—
382C22—241no—
383C22—241no—
384C22C2×#80612yes—
385C22#500111yescharacteristic
386C22#500211yescharacteristic
387C22#500312yes—
388C22#500512yes—
389C22#521311yescharacteristic
390C22#523812yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—84no—
395C4—242no—
396C4—241no—
397C3#915811yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—44no—
401C2—481no—
402C2#6953112yes—
403C2#6953212yes—
404C2#7154311yescharacteristic
405C2#7158611yescharacteristic
406C2#7159111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#381C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC2×#381D3
hypocenterC3#9158
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
1. C2×C3×#381C2semidirect
3. #72126C2semidirect
4. C2×#4451C2semidirect
5. #67612C2semidirect
10. C2×#930C4semidirect
17. #5212C4semidirect
23. C2×#381D3semidirect
54. C3×#95D4semidirect
158. #95#67semidirect
397. C3#9158semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3812
#840322
#721262
C2×#44512
#676122
#91583