Finite Group Databasebeta

← #10693092

Subgroups — #10693092

Order 768.

900 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2405C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic, complemented
3#72130C212yesmaximal
4#71542C211yesmaximal, characteristic, complemented
5C3⋊#1304C212yesmaximal, complemented
6#9496—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#123—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C3×#394—22no—
11C3×#386—21no—
12C3×#385C412yescomplemented
13C3×#379C412yescomplemented
14C3×#379C2211yescharacteristic
15#5247C2212yes—
16#5247C2212yes—
17C2×#834—24no—
18#5212—21no—
19C4×#217—21no—
20#4951—24no—
21#4526—22no—
22#2405D311yescharacteristic
23#2312—31no—
24#1819—31no—
25#1781—32no—
26#1304—32no—
27C25×C3C2×C411yescharacteristic
28C25×C3C2311yescharacteristic
29C22×C3×D4—22no—
30C22×C3×D4—21no—
31C22×C3×D4—21no—
32C22×C3×D4—22no—
33C22×C3×D4—44no—
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C3×#123—44no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4C2×C412yes—
47C2×C3×C22⋊C4D412yes—
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217D414yes—
54C2×C4⋊Dic3—22no—
55C2×C4×Dic3—22no—
56C3×#95—22no—
57#834—48no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60C23×D4C2×D311yescharacteristic
61C2×#123—21no—
62C22×C22⋊C4C2×D311yescharacteristic
63C2×#99—64no—
64#394—22no—
65#388—64no—
66#386—21no—
67#385Dic312yescomplemented
68#380—61no—
69#379—32no—
70#379—32no—
71#379Dic312yescomplemented
72#379C2×D311yescharacteristic
73C4×C22⋊C4—61no—
74#342—62no—
75C24×C3—21no—
76C24×C3—22no—
77C24×C3—42no—
78C24×C3—42no—
79C24×C3—42no—
80C24×C3—42no—
81C24×C3—81no—
82C24×C3—81no—
83C24×C3C22×C411yescharacteristic
84C24×C3C2×D412yes—
85C24×C3C2×D411yescharacteristic
86C24×C3C2×D411yescharacteristic
87C24×C3#4112yes—
88C2×C3×D4—28no—
89C2×C3×D4—24no—
90C2×C3×D4—42no—
91C2×C3×D4—44no—
92C2×C3×D4—42no—
93C2×C3×D4—44no—
94C2×C3×D4—42no—
95C2×C3×D4—44no—
96C2×C3×D4—41no—
97C2×C3×D4—42no—
98C2×C3×D4—41no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—81no—
109C22×C3×C4C22⋊C412yes—
110C22×C3×C4C22⋊C412yes—
111C22×Dic3—24no—
112C22×Dic3—42no—
113C3×C4⋊C4—44no—
114C3×C22⋊C4—24no—
115C3×C22⋊C4—24no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—82no—
120C3×C22⋊C4—81no—
121#217—44no—
122#217—44no—
123#217—44no—
124#217—44no—
125#217—82no—
126#217—82no—
127C4⋊Dic3—82no—
128C4×Dic3—82no—
129C25C2×Dic311yescharacteristic
130C25C22×D311yescharacteristic
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—44no—
136C23×C4C2×Dic311yescharacteristic
137C23×C4#6711yescharacteristic
138C23×C4#6711yescharacteristic
139#123—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—62no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—34no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4C2×Dic312yes—
155C2×C22⋊C4C2×Dic312yes—
156C2×C22⋊C4#6712yes—
157C2×C42—62no—
158#99—128no—
159#95—22no—
160#95—62no—
161#95—62no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—24no—
167C23×C3—22no—
168C23×C3—24no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—41no—
181C23×C3—42no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—82no—
185C23×C3—82no—
186C23×C3—88no—
187C23×C3—88no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3C4×D412yes—
190C23×C3#12011yescharacteristic
191C23×C3#12112yes—
192C23×C3#12311yescharacteristic
193C3×D4—48no—
194C3×D4—44no—
195C3×D4—44no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×Dic3—44no—
213C2×Dic3—44no—
214C2×Dic3—44no—
215C2×Dic3—82no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218C24—21no—
219C24—22no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—81no—
225C24—81no—
226C24D3×D412yes—
227C24#23712yes—
228C24C22×Dic311yescharacteristic
229C24C2×#6711yescharacteristic
230C24C2×#6711yescharacteristic
231C2×D4—28no—
232C2×D4—24no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—42no—
238C2×D4—44no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—64no—
252C22×C4—81no—
253C22×C4—122no—
254C22×C4#21712yes—
255C22×C4#21712yes—
256C4⋊C4—44no—
257C4⋊C4—242no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—82no—
264C22⋊C4—81no—
265C22⋊C4—124no—
266C22⋊C4—124no—
267C22⋊C4—124no—
268C22⋊C4—124no—
269C22⋊C4—242no—
270C22⋊C4—242no—
271C42—242no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—44no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—42no—
289C22×C3—84no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3—84no—
293C22×C3—84no—
294C22×C3—84no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#38611yescharacteristic
298C22×C3#41012yes—
299C22×C3B(4,𝔽2)12yes—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—82no—
304Dic3—84no—
305Dic3—162no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—22no—
312C23—24no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—82no—
329C23—82no—
330C23—88no—
331C23—88no—
332C23D4×Dic312yes—
333C23#93511yescharacteristic
334C23#93711yescharacteristic
335C23#93912yes—
336C23C2×#21711yescharacteristic
337D4—44no—
338D4—48no—
339D4—44no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—124no—
357C2×C4—124no—
358C2×C4—124no—
359C2×C4—242no—
360C2×C4—242no—
361C2×C4—242no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—42no—
365C2×C3—41no—
366C2×C3—42no—
367C2×C3—82no—
368C2×C3#188911yescharacteristic
369C2×C3#189911yescharacteristic
370C2×C3#191311yescharacteristic
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—21no—
383C22—22no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—42no—
388C22—82no—
389C22—84no—
390C22—82no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—81no—
395C22—81no—
396C22#503212yes—
397C22#522011yescharacteristic
398C22#524012yes—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—82no—
403C4—244no—
404C4—482no—
405C3#949611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2#7166911yescharacteristic
413C2#7170511yescharacteristic
414C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#2405C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenter#2405D3
hypocenterC3#9496
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. #82093C2semidirect
4. #71542C2semidirect
5. C3⋊#1304C2semidirect
12. C3×#385C4semidirect
13. C3×#379C4semidirect
67. #385Dic3semidirect
71. #379Dic3semidirect
405. C3#9496semidirect

Maximal subgroups

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

subgroupindex
C3×#24052
#820932
#721302
#715422
C3⋊#13042
#94963