Finite Group Databasebeta

← #10692949

Subgroups — #10692949

Order 768.

934 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 131 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#82403C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3C3×#1799C211yesmaximal, characteristic, complemented
4#69521C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9468—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#937—21no—
10C2×#930—21no—
11C2×C3×#99—21no—
12C3×#380C2212yes—
13#5212C2211yescharacteristic
14C2×#830—21no—
15C2×#830C2212yes—
16D4×C3⋊C8—24no—
17#4991C412yescomplemented
18C2×#809—24no—
19#4960C412yescomplemented
20#4955—21no—
21#4939—21no—
22#4925—21no—
23C3×#342—21no—
24#2972—31no—
25#2408—31no—
26#1799D311yescharacteristic, complemented
27#1786—32no—
28#1289—32no—
29C24×D3—21no—
30C22×C3×D4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12—22no—
35C22×D12—21no—
36C22×D12—22no—
37C22×D12—44no—
38C22×D12C2×C412yes—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×#217—21no—
48#937—42no—
49#937—42no—
50#930—44no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×C4⋊Dic3—21no—
55C22×C3⋊C8—22no—
56C22×C3⋊C8D414yescomplemented
57C3×#99—42no—
58C3×#95—22no—
59C3⋊#95C2×C412yes—
60#830—22no—
61#830—22no—
62#830—22no—
63#809—48no—
64C4⋊(C3⋊C8)—22no—
65C4×C3⋊C8—22no—
66C23×D4—31no—
67C2×#121—61no—
68C2×#120—61no—
69C2×C4×D4C2×D311yescharacteristic
70C8×D4—64no—
71#431—61no—
72C2×#102—64no—
73C2×#99—21no—
74C2×C22⋊C8—32no—
75C2×C22⋊C8—61no—
76#394—32no—
77#392—32no—
78#390—61no—
79#380—31no—
80#380C2×D312yes—
81#379—61no—
82#342—21no—
83C24×C3C22⋊C411yescharacteristic
84C24×C3C2×D411yescharacteristic
85C23×D3—22no—
86C23×D3—44no—
87C23×D3—42no—
88C23×D3—41no—
89C23×D3—41no—
90C23×D3—42no—
91C23×D3—82no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4D812yes—
95C2×C3×D4SD1612yes—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4#4112yes—
108C2×#67—44no—
109C2×#67—44no—
110C2×#67—82no—
111C22×Dic3—22no—
112C2×D12—28no—
113C2×D12—24no—
114C2×D12—24no—
115C2×D12—44no—
116C2×D12—48no—
117C2×D12—44no—
118C2×D12—42no—
119C2×D12—41no—
120C2×D12—41no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—44no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—42no—
126C3×C22⋊C4—42no—
127C3×C42—21no—
128C3×C42—21no—
129#217—41no—
130#217—41no—
131(C2×D3)⋊C4—44no—
132(C2×D3)⋊C4—44no—
133(C2×D3)⋊C4—82no—
134C4⋊Dic3—42no—
135C2×C3⋊C8—24no—
136C2×C3⋊C8—44no—
137C2×C3⋊C8—82no—
138C25—61no—
139C22×D4—32no—
140C22×D4—62no—
141C22×D4—61no—
142C22×D4—62no—
143C22×D4—62no—
144C22×D4—124no—
145C22×D4#6711yescharacteristic
146C23×C4D1211yescharacteristic
147C23×C4C22×D311yescharacteristic
148C22×C8—34no—
149C22×C8—62no—
150#121—124no—
151#120—122no—
152#120—122no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4C4×D311yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4C4×D311yescharacteristic
164C2×C22⋊C4#6711yescharacteristic
165C2×C42D1211yescharacteristic
166C4⋊C8—62no—
167#102—128no—
168#99—42no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—62no—
172C4×C8—62no—
173#95—22no—
174#95—32no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3#10212yescomplemented
182C23×C3C2×C22⋊C411yescharacteristic
183C23×C3#12112yes—
184C23×C3C2×D811yescharacteristic
185C23×C3C2×SD1611yescharacteristic
186C23×C3#13512yes—
187C22×D3—24no—
188C22×D3—42no—
189C22×D3—48no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—84no—
196C22×D3—84no—
197C22×D3—88no—
198C22×D3—84no—
199C22×D3—82no—
200C22×D3—82no—
201C3×D4—24no—
202C3×D4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—81no—
215C2×C3×C4—82no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4#10212yescomplemented
219C2×C3×C4C4×D412yes—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12311yescharacteristic
222#67—88no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226D12—28no—
227D12—44no—
228D12—44no—
229D12—44no—
230C3⋊C8—44no—
231C24—62no—
232C24—124no—
233C24—122no—
234C24—121no—
235C24—121no—
236C24—122no—
237C24—242no—
238C24(C2×D3)⋊C411yescharacteristic
239C24C2×#6711yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—68no—
243C2×D4—64no—
244C2×D4—64no—
245C2×D4—124no—
246C2×D4—122no—
247C2×D4—124no—
248C2×D4—128no—
249C2×D4—124no—
250C2×D4—124no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—242no—
254C2×D4#21312yes—
255C2×D4#21512yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—62no—
262C22×C4(C2×D3)⋊C411yescharacteristic
263C22×C4(C2×D3)⋊C411yescharacteristic
264C22×C4(C2×D3)⋊C411yescharacteristic
265C22×C4C2×C4×D311yescharacteristic
266C22×C4C2×D1211yescharacteristic
267C22×C4D3×D412yes—
268C22×C4#23712yes—
269C2×C8—64no—
270C2×C8—124no—
271C2×C8—242no—
272C4⋊C4—22no—
273C4⋊C4—44no—
274C4⋊C4—122no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—124no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—124no—
283C22⋊C4—242no—
284C42—21no—
285C42—21no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—81no—
298C22×C3#38611yescharacteristic
299C22×C3C2×#10211yescharacteristic
300C22×C3#41611yescharacteristic
301C22×C3#46512yes—
302C22×C3#46612yes—
303C2×D3—44no—
304C2×D3—44no—
305C2×D3—44no—
306C2×D3—82no—
307C2×D3—82no—
308C2×D3—88no—
309C2×D3—88no—
310C2×D3—82no—
311C2×D3—82no—
312C3×C4—22no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—162no—
318C3×C4B(4,𝔽2)12yes—
319Dic3—82no—
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—64no—
327C23—122no—
328C23—128no—
329C23—122no—
330C23—122no—
331C23—122no—
332C23—122no—
333C23—122no—
334C23—244no—
335C23—244no—
336C23—244no—
337C23—248no—
338C23—242no—
339C23—242no—
340C23#80912yescomplemented
341C23C2×(C2×D3)⋊C411yescharacteristic
342C23C2×#21311yescharacteristic
343C23#93912yes—
344C23C2×#21511yescharacteristic
345C23#95012yes—
346D4—24no—
347D4—24no—
348D4—68no—
349D4—124no—
350D4—124no—
351D4—124no—
352D4—248no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4#80912yescomplemented
372C2×C4#67⋊C412yes—
373C2×C4#88211yescharacteristic
374C2×C4#88611yescharacteristic
375C8—124no—
376C2×C3—21no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—42no—
381C2×C3#188911yescharacteristic
382C2×C3#189311yescharacteristic
383C2×C3#191011yescharacteristic
384D3—84no—
385C22—21no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—124no—
398C22—124no—
399C22—124no—
400C22—242no—
401C22—242no—
402C22—248no—
403C22—248no—
404C22—242no—
405C22—242no—
406C22#495811yescharacteristic
407C22C2×#80911yescharacteristic
408C22#500211yescharacteristic
409C22#503812yes—
410C22#504812yes—
411C4—21no—
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—162no—
417C4—242no—
418C4#474112yes—
419C3#946811yescharacteristic, complemented, Sylow
420C2—22no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—42no—
425C2—244no—
426C2#6951511yescharacteristic
427C2#7002211yescharacteristic
428C2#7004711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1799C2
socleC22×C3#386
prime radical#82403C2
cosocleC23×C3×C4C23
hypercenter#1799D3
hypocenterC3#9468
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. C3×#1799C2semidirect
4. #69521C2semidirect
5. #67625C2semidirect
17. #4991C4semidirect
19. #4960C4semidirect
26. #1799D3semidirect
56. C22×C3⋊C8D4semidirect
181. C23×C3#102semidirect
218. C2×C3×C4#102semidirect
340. C23#809semidirect
371. C2×C4#809semidirect
419. C3#9468semidirect

Maximal subgroups

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

subgroupindex
#824032
#823012
C3×#17992
#695212
#676252
#94683