Finite Group Databasebeta

← #10690737

Subgroups — #10690737

Order 768.

560 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 119 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#82326C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4C2×#4469C212yesmaximal, complemented
5#67678C212yesmaximal, complemented
6#9110—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#935C411yescharacteristic
9C22×Dic3⋊C4C2211yescharacteristic
10C3×#435—24no—
11C3×#432—21no—
12C3×#407—21no—
13C2×C3×C22⋊C8C2212yes—
14#5220C411yescharacteristic
15#5210—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2212yes—
18#4976—21no—
19#4946C411yescharacteristic
20#4945—21no—
21#4943C411yescharacteristic
22#4933—21no—
23#4528—21no—
24#4525—21no—
25#4469—24no—
26#2973D311yescharacteristic
27#2322—31no—
28#1459—31no—
29C2×#324—32no—
30#1282—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
48#935—22no—
49#935—22no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—22no—
54C2×Dic3⋊C4—22no—
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C22×C3⋊C8—22no—
58C3×C4⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8D414yes—
62C3×#97—22no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#830—22no—
69#830—22no—
70C2×#123—31no—
71C2×C4×D4C2×D311yescharacteristic
72C22×C4⋊C4—31no—
73#435—24no—
74#432—21no—
75#407—21no—
76C2×C22⋊C8—32no—
77C2×C22⋊C8C2×D312yes—
78#389—61no—
79#388—61no—
80#387—61no—
81#386—31no—
82#385—31no—
83#385—61no—
84#381—31no—
85C4×C22⋊C4—31no—
86#324—61no—
87#324—64no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22×C411yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—81no—
98C22×C3×C4—81no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—22no—
109C22×Dic3—21no—
110C22×Dic3—22no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—81no—
115C22×Dic3M1612yescomplemented
116C3×M16—82no—
117C2×C3×C8—24no—
118C2×C3×C8—82no—
119C3×C4⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C42—21no—
123C3×C42—21no—
124#217—24no—
125#217—21no—
126#217—22no—
127#217—81no—
128#217M1612yescomplemented
129Dic3⋊C4—24no—
130Dic3⋊C4—41no—
131Dic3⋊C4—42no—
132Dic3⋊C4—41no—
133Dic3⋊C4—81no—
134C4×Dic3—81no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—82no—
137C22×D4C4×D311yescharacteristic
138C23×C4—31no—
139C23×C4—31no—
140C23×C4#6711yescharacteristic
141C23×C4C22×D311yescharacteristic
142C2×M16—22no—
143C22×C8—22no—
144C22×C8—62no—
145#123—62no—
146#123—62no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—62no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4D1211yescharacteristic
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4C4×D311yescharacteristic
161C2×C22⋊C4D1211yescharacteristic
162C2×C42—61no—
163C2×C42#6711yescharacteristic
164C4⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169C22⋊C8#6714yes—
170#97—22no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3C4×D412yes—
185C23×C3C2×M1611yescharacteristic
186C23×C3C2×M1611yescharacteristic
187C23×C3#13112yes—
188C3×D4—42no—
189C3×D4—42no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—81no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12112yes—
205C2×C3×C4#12311yescharacteristic
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—24no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—82no—
221C3×C8—44no—
222C3⋊C8—82no—
223C24(C2×D3)⋊C411yescharacteristic
224C24C2×C4×D311yescharacteristic
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—32no—
232C22×C4—61no—
233C22×C4—62no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—241no—
243C22×C4(C2×D3)⋊C411yescharacteristic
244C22×C4(C2×D3)⋊C411yescharacteristic
245C22×C4(C2×D3)⋊C411yescharacteristic
246C22×C4C2×D1211yescharacteristic
247C22×C4#23512yes—
248C22×C4C2×#6711yescharacteristic
249C22×C4C2×#6712yes—
250M16—82no—
251C2×C8—24no—
252C2×C8—82no—
253C2×C8—122no—
254C2×C8—242no—
255C4⋊C4—22no—
256C4⋊C4—64no—
257C4⋊C4—121no—
258C4⋊C4—122no—
259C4⋊C4—121no—
260C4⋊C4—241no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—32no—
264C22⋊C4—64no—
265C22⋊C4—61no—
266C22⋊C4—62no—
267C22⋊C4—241no—
268C42—21no—
269C42—21no—
270C42—241no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—81no—
283C22×C3#38611yescharacteristic
284C22×C3#40711yescharacteristic
285C22×C3#40811yescharacteristic
286C22×C3#43512yes—
287C22×C3#43612yes—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—42no—
295Dic3—42no—
296Dic3—81no—
297Dic3—81no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23D3×M1611yescharacteristic
306C23D3×M1611yescharacteristic
307C23#90712yes—
308C23C2×(C2×D3)⋊C411yescharacteristic
309C23C4×#6712yes—
310D4—42no—
311D4—42no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—64no—
327C2×C4—81no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—242no—
340C2×C4#92911yescharacteristic
341C2×C4#93012yes—
342C2×C4#95311yescharacteristic
343C8—44no—
344C8—242no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—81no—
350C2×C3#187011yescharacteristic
351C2×C3#191311yescharacteristic
352C2×C3#191511yescharacteristic
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22#512611yescharacteristic
366C22#512712yes—
367C22#512812yes—
368C22#512911yescharacteristic
369C22#521311yescharacteristic
370C4—21no—
371C4—22no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—122no—
377C4—122no—
378C4—241no—
379C4—241no—
380C3#911011yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#7165611yescharacteristic
387C2#7199611yescharacteristic
388C2#7203111yescharacteristic
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#9110
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. C2×#4469C2semidirect
5. #67678C2semidirect
115. C22×Dic3M16semidirect
128. #217M16semidirect
380. C3#9110semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823262
#716002
C2×#44692
#676782
#91103