Finite Group Databasebeta

← #10692984

Subgroups — #10692984

Order 768.

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

normal subgroups only

This group has 958 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
1#82125C211yesmaximal, characteristic
2#82083C211yesmaximal, characteristic, complemented
3C3×#1462C211yesmaximal, characteristic, complemented
4#69991C212yesmaximal, complemented
5#67682C212yesmaximal, complemented
6#9475—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#953—21no—
10C22×#217—21no—
11C3×#407C2212yes—
12#5213—21no—
13#5211C2211yescharacteristic
14C4×#217—21no—
15C4×#217C2212yes—
16C2×#825—24no—
17#4960C412yescomplemented
18#4956—22no—
19#4947C412yescomplemented
20Dic3×C22⋊C4—24no—
21C3×#323—22no—
22#2404—31no—
23#2273—31no—
24#1789—32no—
25#1462D311yescharacteristic, complemented
26#1284—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×#67—22no—
32C23×Dic3—22no—
33C2×#235—22no—
34C2×#235—22no—
35C2×#235—44no—
36C22×D12C2×C411yescharacteristic, complemented
37C22×C4×D3—21no—
38C22×Dic6C2×C411yescharacteristic, complemented
39C2×C3×M16—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4C2×C412yes—
43C2×C3×C22⋊C4D412yes—
44#953—44no—
45C2×#217—22no—
46C2×#217—28no—
47C2×#217—22no—
48C2×#217—22no—
49C2×#217—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3D414yescomplemented
53C3×C22⋊C8—22no—
54C3×C22⋊C8—22no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59C3⋊#95C2×C412yescomplemented
60#825—48no—
61C22×#41—31no—
62C2×#120—61no—
63C22×C22⋊C4—61no—
64C22×C22⋊C4C2×D311yescharacteristic
65C2×#104—64no—
66#407C2×D312yes—
67#393—32no—
68#392—32no—
69#386—61no—
70#386—62no—
71#381—31no—
72C4×C22⋊C4—32no—
73C4×C22⋊C4—64no—
74C4×C22⋊C4—61no—
75#323—22no—
76C24×C3—21no—
77C24×C3—24no—
78C24×C3—21no—
79C24×C3—42no—
80C24×C3—81no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C23×D3—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—81no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4#4112yes—
93C2×#67—24no—
94C2×#67—42no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—44no—
98C2×#67—44no—
99C2×#67—82no—
100C22×Dic3—21no—
101C22×Dic3—24no—
102C22×Dic3—28no—
103C22×Dic3—22no—
104C22×Dic3—82no—
105C22×Dic3—82no—
106#235—28no—
107#235—44no—
108#235—44no—
109#235—44no—
110C2×D12—22no—
111C2×D12—42no—
112C2×C4×D3—42no—
113C2×C4×D3—42no—
114C2×C4×D3—42no—
115C2×Dic6—22no—
116C2×Dic6—42no—
117C3×M16—44no—
118C2×C3×C8—42no—
119C3×C22⋊C4—24no—
120C3×C22⋊C4—24no—
121#217—28no—
122#217—28no—
123#217—44no—
124#217—42no—
125(C2×D3)⋊C4—82no—
126C4×Dic3—44no—
127C4×Dic3—82no—
128C25D1211yescharacteristic
129C2×#41—62no—
130C2×#41—62no—
131C2×#41—124no—
132C22×Q8—31no—
133C22×D4—31no—
134C22×D4—61no—
135C22×D4—62no—
136C23×C4—62no—
137C23×C4—61no—
138C23×C4D1211yescharacteristic
139C23×C4C22×D311yescharacteristic
140C2×M16—22no—
141#120—124no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—68no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4C4×D312yes—
151C2×C22⋊C4#6712yes—
152C2×C42—34no—
153C2×C42—62no—
154#104—128no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157#95—32no—
158#95—62no—
159#95—62no—
160#95—62no—
161#95—61no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—24no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—41no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3—81no—
176C23×C3—88no—
177C23×C3#10414yescomplemented
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3#12011yescharacteristic
180C23×C3#12311yescharacteristic
181C22×D3—22no—
182C22×D3—42no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—42no—
186C2×C3×C4—24no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—81no—
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12112yes—
196#67—44no—
197#67—44no—
198#67—88no—
199C2×Dic3—28no—
200C2×Dic3—22no—
201C2×Dic3—44no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—44no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—88no—
209D12—24no—
210D12—42no—
211C4×D3—48no—
212Dic6—24no—
213Dic6—42no—
214C3×C8—82no—
215C24—21no—
216C24—24no—
217C24—21no—
218C24—42no—
219C24—61no—
220C24—81no—
221C24(C2×D3)⋊C412yes—
222C24C2×D1211yescharacteristic
223#41—68no—
224#41—124no—
225#41—124no—
226#41—124no—
227C2×Q8—62no—
228C2×Q8—122no—
229C2×D4—64no—
230C2×D4—62no—
231C2×D4—122no—
232C2×D4—122no—
233C2×D4—121no—
234C2×D4—121no—
235C2×D4—124no—
236C2×D4—124no—
237C2×D4—242no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—64no—
242C22×C4—61no—
243C22×C4—68no—
244C22×C4—62no—
245C22×C4—81no—
246C22×C4—122no—
247C22×C4—122no—
248C22×C4—122no—
249C22×C4—242no—
250C22×C4—242no—
251C22×C4(C2×D3)⋊C412yes—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D412yes—
254C22×C4#23712yes—
255C22×C4C2×#6711yescharacteristic
256M16—44no—
257C2×C8—42no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—68no—
261C22⋊C4—68no—
262C22⋊C4—124no—
263C22⋊C4—122no—
264C22⋊C4—242no—
265C42—124no—
266C42—242no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—82no—
282C22×C3—84no—
283C22×C3#38611yescharacteristic
284C22×C3C2×#10412yes—
285C22×C3B(4,𝔽2)12yes—
286C2×D3—44no—
287C2×D3—42no—
288C2×D3—82no—
289C2×D3—81no—
290C2×D3—81no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—42no—
294Dic3—48no—
295Dic3—42no—
296Dic3—82no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—41no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—62no—
310C23—81no—
311C23—81no—
312C23—88no—
313C23—122no—
314C23—121no—
315C23—122no—
316C23—121no—
317C23#82514yescomplemented
318C23#88211yescharacteristic
319C23#88611yescharacteristic
320C23C2×(C2×D3)⋊C411yescharacteristic
321Q8—64no—
322Q8—122no—
323D4—64no—
324D4—122no—
325D4—124no—
326D4—124no—
327D4—248no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—68no—
336C2×C4—62no—
337C2×C4—81no—
338C2×C4—128no—
339C2×C4—124no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—124no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—248no—
347C2×C4#67⋊C412yes—
348C2×C4#93912yes—
349C8—82no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—41no—
353C2×C3—44no—
354C2×C3#188911yescharacteristic
355C2×C3#189611yescharacteristic
356C2×C3#191111yescharacteristic
357D3—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—42no—
371C22—42no—
372C22—82no—
373C22—84no—
374C22—124no—
375C22—122no—
376C22—242no—
377C22—241no—
378C22—241no—
379C22#474112yes—
380C22#495811yescharacteristic
381C22C2×#82512yes—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—128no—
386C4—122no—
387C4—242no—
388C3#947511yescharacteristic, complemented, Sylow
389C2—22no—
390C2—22no—
391C2—41no—
392C2—44no—
393C2—242no—
394C2#6996511yescharacteristic
395C2#7002211yescharacteristic
396C2#7007111yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1462C2
socleC22×C3#386
prime radical#82125C2
cosocleC23×C3×C4C23
hypercenter#1462D3
hypocenterC3#9475
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #82083C2semidirect
3. C3×#1462C2semidirect
4. #69991C2semidirect
5. #67682C2semidirect
17. #4960C4semidirect
19. #4947C4semidirect
25. #1462D3semidirect
36. C22×D12C2×C4semidirect
38. C22×Dic6C2×C4semidirect
52. C2×C4×Dic3D4semidirect
59. C3⋊#95C2×C4semidirect
177. C23×C3#104semidirect
317. C23#825semidirect
388. C3#9475semidirect

Maximal subgroups

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

subgroupindex
#821252
#820832
C3×#14622
#699912
#676822
#94753