Finite Group Databasebeta

← #10693172

Subgroups — #10693172

Order 768.

558 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2482C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic, complemented
3#71589C211yesmaximal, characteristic, complemented
4#71539C212yesmaximal
5#67629C212yesmaximal, complemented
6#9512—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C3×#399—21no—
10C3×#397—21no—
11C3×#388—21no—
12C3×#388—21no—
13C3×#385C411yescharacteristic
14C3×#385C411yescharacteristic
15C3×#381C2211yescharacteristic
16C3×#379C411yescharacteristic
17C3×#379C411yescharacteristic
18#5207C2212yes—
19#5207C2212yes—
20#5000—24no—
21#4924—21no—
22C4×#208—21no—
23#4530—21no—
24#4530—21no—
25#4525—24no—
26#2921—31no—
27#2482D311yescharacteristic
28#1822—31no—
29#1462—32no—
30#1306—32no—
31C25×C3C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4C2×C411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×#208—22no—
55C2×#208—22no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—22no—
61#830—22no—
62#830—22no—
63#830D414yes—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66C4×C3⋊C8—22no—
67C2×#117C2×D311yescharacteristic
68C22×C22⋊C4C2×D311yescharacteristic
69#433—64no—
70#423—61no—
71#407—32no—
72#407—32no—
73C4×M16—61no—
74#399—21no—
75#397—21no—
76#388—21no—
77#388—21no—
78#385Dic311yescharacteristic
79#385Dic311yescharacteristic
80#381C2×D311yescharacteristic
81#379Dic311yescharacteristic
82#379Dic311yescharacteristic
83#343—61no—
84#343—61no—
85#323—64no—
86C24×C3—21no—
87C24×C3—22no—
88C24×C3—42no—
89C24×C3—41no—
90C24×C3—41no—
91C24×C3—81no—
92C24×C3C22×C411yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—81no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—81no—
111C3×C4⋊C4—81no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—81no—
122C3×C22⋊C4—81no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42—21no—
127#208—42no—
128#208—42no—
129#208—42no—
130#208—42no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—42no—
133C25C2×Dic311yescharacteristic
134C23×C4C2×Dic311yescharacteristic
135C23×C4C22×D311yescharacteristic
136C2×M16—62no—
137C2×M16—62no—
138#117—22no—
139#117—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4#6711yescharacteristic
143C2×C4⋊C4#6711yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4C2×Dic311yescharacteristic
153C2×C22⋊C4C2×Dic311yescharacteristic
154C2×C22⋊C4C2×Dic311yescharacteristic
155C2×C22⋊C4C2×Dic311yescharacteristic
156C2×C42#6711yescharacteristic
157C2×C42#6711yescharacteristic
158C4⋊C8—64no—
159C22⋊C8—34no—
160C22⋊C8—62no—
161C22⋊C8—62no—
162C22⋊C8—62no—
163#97—62no—
164C4×C8—62no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—22no—
172C23×C3—24no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—82no—
183C23×C3—82no—
184C23×C3—84no—
185C23×C3—84no—
186C23×C3C2×C22⋊C411yescharacteristic
187C23×C3C4×D412yes—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—82no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12112yes—
214C2×C3×C4#12311yescharacteristic
215C3⋊C8—44no—
216C3⋊C8—82no—
217C24—21no—
218C24—22no—
219C24—42no—
220C24—41no—
221C24—41no—
222C24—81no—
223C24C22×Dic311yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—81no—
231C22×C4#21711yescharacteristic
232C22×C4#21711yescharacteristic
233C22×C4#21711yescharacteristic
234C22×C4#21711yescharacteristic
235C22×C4D3×D412yes—
236C22×C4#23712yes—
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—122no—
240M16—122no—
241M16—122no—
242M16—122no—
243C2×C8—64no—
244C2×C8—122no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—84no—
277C22×C3—84no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3#38611yescharacteristic
281C22×C3#41011yescharacteristic
282C22×C3#41011yescharacteristic
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4—81no—
293C23—21no—
294C23—21no—
295C23—22no—
296C23—24no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—82no—
307C23—82no—
308C23—84no—
309C23—84no—
310C23D4×Dic312yes—
311C23C2×#21711yescharacteristic
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4#93511yescharacteristic
337C2×C4#93711yescharacteristic
338C2×C4#93912yes—
339C8—124no—
340C8—242no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—42no—
344C2×C3—82no—
345C2×C3#189111yescharacteristic
346C2×C3#189911yescharacteristic
347C2×C3#191411yescharacteristic
348C22—21no—
349C22—21no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—84no—
361C22—84no—
362C22—81no—
363C22—81no—
364C22#522011yescharacteristic
365C22#524011yescharacteristic
366C22#524011yescharacteristic
367C4—21no—
368C4—22no—
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—81no—
377C3#951211yescharacteristic, complemented, Sylow
378C2—21no—
379C2—21no—
380C2—42no—
381C2—82no—
382C2#7170511yescharacteristic
383C2#7171411yescharacteristic
384C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2482C2
socleC22×C3#386
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenter#2482D3
hypocenterC3#9512
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #82209C2semidirect
3. #71589C2semidirect
5. #67629C2semidirect
377. C3#9512semidirect

Maximal subgroups

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

subgroupindex
C3×#24822
#822092
#715892
#715392
#676292
#95123