Finite Group Databasebeta

← #10692907

Subgroups — #10692907

Order 768.

904 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2435C211yesmaximal, characteristic
2#84043C211yesmaximal, characteristic
3#72129C211yesmaximal, characteristic
4#71689C212yesmaximal, complemented
5#67616C212yesmaximal, complemented
6#9459—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#124C412yescomplemented
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#217—21no—
11C3×#394—22no—
12C3×#390C412yescomplemented
13C3×#379—21no—
14C3×#379—21no—
15C3×C4×C22⋊C4C2211yescharacteristic
16#5247—21no—
17#5220—24no—
18C2×#833—22no—
19#5211C2212yes—
20#5207C2212yes—
21#4538—24no—
22#2435D311yescharacteristic
23#2308—31no—
24#1783—31no—
25#1511—32no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C25×C3D411yescharacteristic
29C22×C3×Q8C2×C411yescharacteristic, complemented
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—44no—
33C22×C3×D4C2×C412yes—
34C22×C3×D4D412yes—
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—22no—
37C3×#124—44no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C42C2×C412yescomplemented
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—44no—
54C2×#208—22no—
55C3×#95—21no—
56#833—44no—
57C3⋊#95—22no—
58C3⋊#95D414yes—
59#830—22no—
60#830—22no—
61C23×D4C2×D311yescharacteristic
62C2×#124Dic312yescomplemented
63C2×C22⋊Q8C2×D311yescharacteristic
64C22×C22⋊C4—61no—
65C2×#100—62no—
66#407—32no—
67#394—22no—
68#390Dic312yescomplemented
69#386—64no—
70#381—32no—
71#379—21no—
72#379—21no—
73#379—61no—
74C4×C22⋊C4C2×D311yescharacteristic
75#328—64no—
76C24×C3—21no—
77C24×C3—24no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—21no—
81C24×C3—42no—
82C24×C3—42no—
83C24×C3—42no—
84C24×C3—81no—
85C24×C3—81no—
86C24×C3C22⋊C412yes—
87C24×C3C22⋊C412yes—
88C24×C3C2×D411yescharacteristic
89C2×C3×Q8—22no—
90C2×C3×Q8—42no—
91C2×C3×D4—24no—
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—24no—
95C2×C3×D4—42no—
96C2×C3×D4—42no—
97C2×C3×D4—44no—
98C2×C3×D4—44no—
99C2×C3×D4—48no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—42no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C22×Dic3—24no—
109C22×Dic3—44no—
110C22×Dic3—82no—
111C22×Dic3—82no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—42no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—44no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—82no—
120C3×C22⋊C4—82no—
121C3×C42—42no—
122#217—44no—
123#217—44no—
124#217—82no—
125#217—82no—
126#208—44no—
127C2×C3⋊C8—42no—
128C25#6711yescharacteristic
129C25#6711yescharacteristic
130C22×Q8C2×Dic311yescharacteristic, complemented
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—44no—
134C22×D4C2×Dic312yes—
135C22×D4#6712yes—
136C23×C4—62no—
137C23×C4C22×D311yescharacteristic
138C2×M16—62no—
139#124—44no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×Dic311yescharacteristic, complemented
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—124no—
155C2×C42C2×Dic312yescomplemented
156#100—124no—
157C22⋊C8—62no—
158C22⋊C8—62no—
159#95—21no—
160#95—34no—
161#95—62no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—24no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—41no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—42no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3—81no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3—81no—
187C23×C3—88no—
188C23×C3—88no—
189C23×C3#9914yes—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3#12311yescharacteristic
193C3×Q8—82no—
194C3×D4—48no—
195C3×D4—48no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4C4×D412yes—
208C2×C3×C4#12112yes—
209C2×Dic3—44no—
210C2×Dic3—44no—
211C2×Dic3—44no—
212C2×Dic3—88no—
213C3⋊C8—82no—
214C24—21no—
215C24—24no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—81no—
223C24—81no—
224C24#21712yes—
225C24#21712yes—
226C24C2×#6711yescharacteristic
227C2×Q8—22no—
228C2×Q8—42no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—48no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—64no—
243C22×C4—124no—
244C22×C4—242no—
245C22×C4—242no—
246C22×C4D3×D412yes—
247C22×C4#23712yes—
248C22×C4C22×Dic311yescharacteristic
249C22×C4C2×#6711yescharacteristic
250M16—124no—
251C2×C8—122no—
252C4⋊C4—22no—
253C4⋊C4—42no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—124no—
262C22⋊C4—124no—
263C22⋊C4—242no—
264C22⋊C4—242no—
265C42—42no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—44no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—84no—
287C22×C3—84no—
288C22×C3—84no—
289C22×C3#38611yescharacteristic
290C22×C3C2×#9912yes—
291C22×C3#45512yes—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—82no—
295C3×C4—81no—
296C3×C4—81no—
297Dic3—84no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—88no—
324C23—88no—
325C23#83414yes—
326C23#93511yescharacteristic
327C23#93711yescharacteristic
328C23C2×#21711yescharacteristic
329Q8—82no—
330D4—48no—
331D4—48no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—124no—
344C2×C4—124no—
345C2×C4—124no—
346C2×C4—248no—
347C2×C4D4×Dic312yes—
348C2×C4#93912yes—
349C8—242no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—41no—
353C2×C3—44no—
354C2×C3—82no—
355C2×C3#188511yescharacteristic
356C2×C3#189811yescharacteristic
357C2×C3#191111yescharacteristic
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—82no—
377C22—82no—
378C22—84no—
379C22—84no—
380C22—84no—
381C22#506112yes—
382C22C2×#83412yes—
383C22#522011yescharacteristic
384C4—22no—
385C4—22no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—244no—
390C3#945911yescharacteristic, complemented, Sylow
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2#7167011yescharacteristic
397C2#7172311yescharacteristic
398C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2435C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenter#2435D3
hypocenterC3#9459
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
4. #71689C2semidirect
5. #67616C2semidirect
8. C2×C3×#124C4semidirect
12. C3×#390C4semidirect
29. C22×C3×Q8C2×C4semidirect
41. C2×C3×C4⋊C4C2×C4semidirect
48. C2×C3×C42C2×C4semidirect
62. C2×#124Dic3semidirect
68. #390Dic3semidirect
130. C22×Q8C2×Dic3semidirect
143. C2×C4⋊C4C2×Dic3semidirect
155. C2×C42C2×Dic3semidirect
390. C3#9459semidirect

Maximal subgroups

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

subgroupindex
C3×#24352
#840432
#721292
#716892
#676162
#94593