Finite Group Databasebeta

← #10693116

Subgroups — #10693116

Order 768.

928 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2309C211yesmaximal, characteristic
2#84047C211yesmaximal, characteristic
3#72130C211yesmaximal, characteristic
4#69495C212yesmaximal, complemented
5C3⋊#1304C212yesmaximal, complemented
6#9501—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#117—21no—
9C22×#217C2211yescharacteristic
10C2×#935C412yescomplemented
11C3×#390—24no—
12C3×#379—21no—
13C3×#379C2212yes—
14#5247—21no—
15#5247C2212yes—
16#5247C2211yescharacteristic
17#5222—22no—
18#5220C412yescomplemented
19C2×#834—22no—
20#5210—21no—
21#4527—24no—
22#2405—31no—
23#2309D311yescharacteristic
24#1819—32no—
25#1781—31no—
26#1304—32no—
27C25×C3D411yescharacteristic
28C25×C3C2311yescharacteristic
29C22×C3×D4—22no—
30C22×C3×D4—22no—
31C22×C3×D4—44no—
32C22×C3×D4C2×C412yes—
33C22×C3×D4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×Dic3C2×C411yescharacteristic
36C23×Dic3C2×C411yescharacteristic
37C3×#117—44no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4D414yes—
44C2×C3×C42—22no—
45C2×#217—22no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—21no—
55C2×#217—22no—
56C2×#217C2×C412yes—
57#935—44no—
58C2×Dic3⋊C4—22no—
59#834—44no—
60C3⋊#95—22no—
61C23×D4C2×D311yescharacteristic
62C2×#123—32no—
63C2×#117—21no—
64C22×C22⋊C4—31no—
65C2×#99—62no—
66#394—62no—
67#390—24no—
68#386—32no—
69#385—61no—
70#379—21no—
71#379—31no—
72#379—32no—
73#379—61no—
74#379C2×D312yes—
75#342—64no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—42no—
79C24×C3—42no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3—81no—
83C24×C3—81no—
84C24×C3C22⋊C412yes—
85C24×C3C22×C411yescharacteristic
86C24×C3C2×D411yescharacteristic
87C24×C3C2×D411yescharacteristic
88C24×C3C2×D412yes—
89C24×C3#4112yes—
90C2×C3×D4—24no—
91C2×C3×D4—24no—
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—42no—
95C2×C3×D4—42no—
96C2×C3×D4—44no—
97C2×C3×D4—44no—
98C2×C3×D4—48no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4C22⋊C412yes—
104C22×Dic3—21no—
105C22×Dic3—22no—
106C22×Dic3—21no—
107C22×Dic3—22no—
108C22×Dic3—22no—
109C22×Dic3—42no—
110C22×Dic3—42no—
111C22×Dic3—81no—
112C3×C4⋊C4—44no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—82no—
116C3×C22⋊C4—82no—
117C3×C22⋊C4—82no—
118C3×C42—44no—
119#217—44no—
120#217—42no—
121#217—41no—
122#217—42no—
123#217—41no—
124#217—42no—
125#217—44no—
126#217—44no—
127#217—81no—
128#217—82no—
129#217—82no—
130Dic3⋊C4—44no—
131C25#6711yescharacteristic
132C25C22×D311yescharacteristic
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4C4×D312yes—
137C22×D4D1212yes—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4#6711yescharacteristic
141#123—124no—
142#117—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—62no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—32no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—62no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4#6714yes—
162C2×C42—22no—
163#99—124no—
164#95—62no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—24no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—42no—
181C23×C3—41no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—81no—
190C23×C3—81no—
191C23×C3—88no—
192C23×C3—88no—
193C23×C3C2×C22⋊C411yescharacteristic
194C23×C3C4×D412yes—
195C23×C3#12011yescharacteristic
196C23×C3#12112yes—
197C23×C3#12311yescharacteristic
198C3×D4—48no—
199C3×D4—48no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4#9914yes—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218C2×Dic3—82no—
219C2×Dic3—82no—
220C2×Dic3—82no—
221C24—21no—
222C24—21no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—81no—
228C24—81no—
229C24(C2×D3)⋊C412yes—
230C24C2×C4×D311yescharacteristic
231C24C2×D1211yescharacteristic
232C24#23512yes—
233C24C2×#6711yescharacteristic
234C24C2×#6712yes—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—48no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4—241no—
256C22×C4(C2×D3)⋊C412yes—
257C4⋊C4—44no—
258C4⋊C4—124no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—124no—
265C22⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—122no—
268C22⋊C4—121no—
269C22⋊C4—122no—
270C22⋊C4—124no—
271C22⋊C4—124no—
272C22⋊C4—241no—
273C22⋊C4—242no—
274C22⋊C4—242no—
275C42—44no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—82no—
296C22×C3—84no—
297C22×C3—82no—
298C22×C3—84no—
299C22×C3—84no—
300C22×C3—84no—
301C22×C3#38611yescharacteristic
302C22×C3C2×#9912yes—
303C22×C3#45812yes—
304C3×C4—24no—
305C3×C4—84no—
306Dic3—82no—
307Dic3—81no—
308Dic3—81no—
309Dic3—162no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—88no—
337C23—88no—
338C23C2×(C2×D3)⋊C411yescharacteristic
339C23C4×#6712yes—
340C23#92911yescharacteristic
341C23#93012yes—
342C23#95311yescharacteristic
343D4—48no—
344D4—48no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—242no—
361C2×C4—242no—
362C2×C4—242no—
363C2×C4—242no—
364C2×C4—242no—
365C2×C4#80614yes—
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—42no—
369C2×C3—42no—
370C2×C3—41no—
371C2×C3—82no—
372C2×C3#189011yescharacteristic
373C2×C3#189811yescharacteristic
374C2×C3#191311yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—21no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—84no—
395C22—82no—
396C22—82no—
397C22—84no—
398C22—84no—
399C22—84no—
400C22C2×#80612yes—
401C22#521311yescharacteristic
402C22#524312yes—
403C4—24no—
404C4—84no—
405C4—242no—
406C4—241no—
407C4—241no—
408C4—482no—
409C3#950111yescharacteristic, complemented, Sylow
410C2—22no—
411C2—22no—
412C2—42no—
413C2—41no—
414C2—42no—
415C2—82no—
416C2#7154311yescharacteristic
417C2#7165511yescharacteristic
418C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#2309C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenter#2309D3
hypocenterC3#9501
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2309

Complements

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

NH ≅ G/Ntype
4. #69495C2semidirect
5. C3⋊#1304C2semidirect
10. C2×#935C4semidirect
18. #5220C4semidirect
409. C3#9501semidirect

Maximal subgroups

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

subgroupindex
C3×#23092
#840472
#721302
#694952
C3⋊#13042
#95013