Finite Group Databasebeta

← #10897648

Subgroups — #10897648

Order 768.

662 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 141 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#84000C211yesmaximal, characteristic, complemented
2#79039C211yesmaximal, characteristic, complemented
3C3×#2050C211yesmaximal, characteristic, complemented
4#72118C211yesmaximal, characteristic
5#70526C211yesmaximal, characteristic, complemented
6#70383C212yesmaximal, complemented
7#70375C212yesmaximal, complemented
8#70136C212yesmaximal, complemented
9#70130C212yesmaximal, complemented
10#68721C211yesmaximal, characteristic, complemented
11#68716C211yesmaximal, characteristic, complemented
12#29385—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14#5828C2212yes—
15#5819C2212yes—
16C2×#872C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C2×C3×#104C2212yes—
19C3×#413C2212yes—
20#5236C2212yes—
21C2×#837C2212yes—
22#5197—22no—
23#5196—22no—
24#5126C2211yescharacteristic
25#5126C2211yescharacteristic
26#5126C2212yes—
27#5125C2211yescharacteristic
28#5125C2212yes—
29Dic3×M16C2211yescharacteristic
30Dic3×M16C2211yescharacteristic
31#4924C2211yescharacteristic
32#4840—22no—
33#4840—22no—
34C3×#338C2211yescharacteristic
35#4559C2212yes—
36#4559C2212yes—
37#4557C2211yescharacteristic
38#4555C2212yes—
39#4523C2212yes—
40#2956—31no—
41#2921—31no—
42#2102—31no—
43#2091—31no—
44#2050D311yescharacteristic, complemented
45#2010—31no—
46#1932—32no—
47#1897—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C22×C4×D3C2311yescharacteristic
53C3×#137—22no—
54C3×AGL(1,ℤ/8)—22no—
55C3×#135—22no—
56C2×C3×M16C2312yes—
57C2×C3×M16C2311yescharacteristic
58C2×C3×M16C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60#948—24no—
61#948—42no—
62#946—22no—
63#938—22no—
64#937—22no—
65#935—22no—
66C4×#67—42no—
67C2×(C2×D3)⋊C4—21no—
68#926C2312yes—
69C2×Dic3⋊C4—21no—
70C2×C4×Dic3C2311yescharacteristic
71C2×#208—42no—
72C2×#208C2311yescharacteristic
73C2×#208C2312yes—
74C2×#208C2311yescharacteristic
75#872—22no—
76#872—21no—
77#872—21no—
78C3×C4⋊C8—22no—
79C3×#104—24no—
80C3×#101—22no—
81C3×#100—22no—
82#837—22no—
83#837—22no—
84#837—42no—
85(C2×D3)⋊C8—22no—
86(C2×D3)⋊C8—22no—
87(C2×D3)⋊C8—42no—
88#819—22no—
89#819—42no—
90#815—21no—
91#815D412yescomplemented
92C8×Dic3—21no—
93C8×Dic3D412yescomplemented
94C4⋊(C3⋊C8)—22no—
95#575C2×D311yescharacteristic
96#568—32no—
97#525—32no—
98C2×#117—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#433—62no—
103#430—31no—
104#430—32no—
105#423—31no—
106#423C2×D311yescharacteristic
107#421—32no—
108C2×#104—32no—
109C2×#104C2×D312yes—
110#413C2×D312yes—
111#407—31no—
112#407—31no—
113#407—32no—
114C4×M16—31no—
115C4×M16—31no—
116#344—32no—
117#344—32no—
118#339—32no—
119#339—32no—
120#338C2×D311yescharacteristic
121H(3,ℤ/4)—31no—
122C23×D3#4111yescharacteristic
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—22no—
127C2×C3×Q8#4112yes—
128C2×C3×D4—22no—
129C2×C3×D4#4112yes—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4C2411yescharacteristic
133C2×#67—42no—
134C22×Dic3—21no—
135C22×Dic3#4111yescharacteristic
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—22no—
139C2×C4×D3—22no—
140C3×Q16—22no—
141C3×SD16—24no—
142C3×D8—22no—
143C3×M16—22no—
144C3×M16—22no—
145C3×M16—22no—
146C3×M16—22no—
147C2×C3×C8—21no—
148C2×C3×C8—22no—
149C2×C3×C8C2×D412yes—
150C3×C42—21no—
151C3×C42#4112yes—
152#217—22no—
153#217—42no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—22no—
156(C2×D3)⋊C4—42no—
157C4⋊Dic3—22no—
158Dic3⋊C4—22no—
159Dic3⋊C4—22no—
160Dic3⋊C4—42no—
161C4×Dic3—21no—
162C4×Dic3—22no—
163C4×Dic3C2×D411yescharacteristic
164C4×Dic3C2×D411yescharacteristic
165#208—22no—
166#208—22no—
167#208—22no—
168#208—22no—
169#208—44no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—44no—
173C2×C3⋊C8C2×D412yes—
174C2×#41C22×D312yes—
175C23×C4—31no—
176#137—22no—
177AGL(1,ℤ/8)—22no—
178#135—22no—
179#131—64no—
180#131—122no—
181C2×M16—31no—
182C2×M16—32no—
183C2×M16—31no—
184C2×M16—122no—
185C2×M16C22×D312yes—
186C2×M16C22×D311yescharacteristic
187C2×M16C22×D311yescharacteristic
188#123—62no—
189C22⋊Q8—62no—
190#121—62no—
191#120—62no—
192C4×D4—122no—
193#117—32no—
194#117—61no—
195#117—61no—
196#117—62no—
197C2×C4⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—31no—
200C2×C42C22×D311yescharacteristic
201#108—62no—
202#108—122no—
203C4⋊C8—22no—
204C4⋊C8—62no—
205#104—24no—
206#104—62no—
207#104—62no—
208#104—122no—
209#101—22no—
210#100—22no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—32no—
215#97—61no—
216C4×C8—32no—
217C4×C8—61no—
218C23×C3—22no—
219C23×C3C2×#4111yescharacteristic
220C22×D3—21no—
221C22×D3—22no—
222C22×D3—81no—
223C3×Q8—22no—
224C3×Q8—22no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—42no—
228C3×D4—42no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4C22×D411yescharacteristic
235C2×C3×C4C2×#4112yes—
236C2×C3×C4C2×#4111yescharacteristic
237C2×C3×C4H(5,𝔽2)12yes—
238#67—84no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—81no—
245C4×D3—41no—
246C4×D3—41no—
247C4×D3—42no—
248C3×C8—22no—
249C3×C8—41no—
250C3×C8—42no—
251C3⋊C8—22no—
252C3⋊C8—42no—
253C3⋊C8—41no—
254C24—31no—
255#41—22no—
256#41—22no—
257#41—22no—
258#41—22no—
259C2×Q8#23712yes—
260C2×D4—22no—
261C2×D4—122no—
262C2×D4#23712yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—31no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4C23×D311yescharacteristic
272Q16—22no—
273SD16—24no—
274D8—22no—
275M16—22no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—62no—
283M16—124no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—62no—
289C2×C8—124no—
290C2×C8D3×D412yes—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—122no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C42—21no—
301C42—31no—
302C42—31no—
303C42—61no—
304C42—62no—
305C42#23712yes—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3#52511yescharacteristic
310C22×C3#55112yes—
311C2×D3—42no—
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—41no—
315C3×C4—42no—
316C3×C4—42no—
317C3×C4#52511yescharacteristic
318C3×C4#53511yescharacteristic
319C3×C4#55312yes—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323Dic3—82no—
324C23—22no—
325C23—62no—
326C23—61no—
327C23—241no—
328C23D3×#4111yescharacteristic
329Q8—22no—
330Q8—22no—
331D4—22no—
332D4—22no—
333D4—42no—
334D4—42no—
335D4—244no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—81no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—241no—
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×#23712yes—
352C2×C4C2×#23711yescharacteristic
353C2×C4#100912yes—
354C8—22no—
355C8—41no—
356C8—42no—
357C8—62no—
358C8—122no—
359C8—121no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—82no—
363C2×C3#259811yescharacteristic
364D3—81no—
365C22—41no—
366C22—42no—
367C22—82no—
368C22—122no—
369C22—122no—
370C22—122no—
371C22—121no—
372C22#560611yescharacteristic
373C22#565812yes—
374C4—42no—
375C4—42no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4—242no—
380C4#558611yescharacteristic
381C4#561812yes—
382C4#567411yescharacteristic
383C3#2938511yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—82no—
387C2—241no—
388C2#7696211yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2050C2
socleC2×C3#2598
prime radical#72118C2
hypercenter#2050D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2050

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #79039C2semidirect
3. C3×#2050C2semidirect
5. #70526C2semidirect
6. #70383C2semidirect
7. #70375C2semidirect
8. #70136C2semidirect
9. #70130C2semidirect
10. #68721C2semidirect
11. #68716C2semidirect
44. #2050D3semidirect
91. #815D4semidirect
93. C8×Dic3D4semidirect
383. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#840002
#790392
C3×#20502
#721182
#705262
#703832
#703752
#701362
#701302
#687212
#687162
#293853