Finite Group Databasebeta

← #10826744

Subgroups — #10826744

Order 768.

1764 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 199 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#87273C212yesmaximal, complemented
2#78686C214yesmaximal, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#70024C214yesmaximal, complemented
5#69539C214yesmaximal, complemented
6#23896—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#1004C2212yescomplemented
9#5829—28no—
10#5824—28no—
11#5816—22no—
12#5804—22no—
13#5804C2214yes—
14#5798C2212yescomplemented
15#5791—22no—
16C2×#929—22no—
17C2×#929C2214yescomplemented
18#5493C2214yes—
19C2×C3×#99C2214yescomplemented
20C3×#381C2212yescomplemented
21C2×#806C2218yescomplemented
22#4742—28no—
23#4741—28no—
24#4526C2214yescomplemented
25#3485—32no—
26#3026—34no—
27#2022—34no—
28#1913—34no—
29#1782D311yescharacteristic, complemented
30C2×C3×#41—28no—
31C2×C3×#41—42no—
32C22×C3×Q8D411yescharacteristic, complemented
33C22×C3×D4D411yescharacteristic, complemented
34C22×C3×D4C2312yes—
35C23×C3×C4D412yescomplemented
36C23×C3×C4C2311yescharacteristic
37C22×#67—22no—
38C22×#67C2314yes—
39#1004—28no—
40#1004—44no—
41C2×#237—24no—
42C2×D3×D4—24no—
43C2×#235—22no—
44C2×C3×C22⋊C4C2314yes—
45#953—24no—
46C2×#217—22no—
47C2×#217C2314yes—
48#947—24no—
49#946—28no—
50#944—24no—
51#939—28no—
52#939—28no—
53#937—24no—
54#937D418yescomplemented
55#935—24no—
56#935D418yescomplemented
57D4×Dic3—28no—
58#930—24no—
59#930—44no—
60#929—28no—
61#929—24no—
62#929—24no—
63#929—24no—
64#929—28no—
65C4×#67—28no—
66C2×(C2×D3)⋊C4—24no—
67C2×(C2×D3)⋊C4—24no—
68#926—22no—
69#924—24no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—24no—
72#886—28no—
73#882—28no—
74C3×#99—28no—
75C3×#95—42no—
76#806—216no—
77C2×H(5,𝔽2)—32no—
78C22×#41C2×D311yescharacteristic
79#547—68no—
80#546—68no—
81#536—62no—
82#534—34no—
83#534—34no—
84#534—62no—
85#525—62no—
86C2×#123—34no—
87C2×#123—62no—
88#518—32no—
89#458—68no—
90B(4,𝔽2)—68no—
91C2×#99—38no—
92C2×#99C2×D314yescomplemented
93#381C2×D312yescomplemented
94#342—34no—
95C24×C3C2×D412yes—
96C24×C3C2411yescharacteristic
97C23×D3—22no—
98C23×D3C2×D414yes—
99C3×#41—416no—
100C2×C3×Q8—24no—
101C2×C3×Q8—41no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—24no—
105C2×C3×D4—41no—
106C2×C3×D4C2×D418yes—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—22no—
110C22×C3×C4—24no—
111C22×C3×C4—24no—
112C22×C3×C4—42no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D412yes—
116C2×#67—24no—
117C2×#67—28no—
118C2×#67—24no—
119C2×#67—24no—
120C2×#67—28no—
121C2×#67—28no—
122C2×#67—44no—
123C2×#67—44no—
124C22×Dic3—22no—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3C2×D414yes—
128#237—416no—
129D3×D4—416no—
130#235—44no—
131#235—44no—
132C2×D12—22no—
133C2×D12—44no—
134C2×C4×D3—24no—
135C2×Dic6—22no—
136C3×C22⋊C4—28no—
137#217—24no—
138#217—28no—
139#217—24no—
140#217C2×D418yes—
141(C2×D3)⋊C4—28no—
142(C2×D3)⋊C4—28no—
143(C2×D3)⋊C4—28no—
144(C2×D3)⋊C4—48no—
145C4⋊Dic3—24no—
146C4⋊Dic3—44no—
147Dic3⋊C4—28no—
148Dic3⋊C4—28no—
149Dic3⋊C4—28no—
150C4×Dic3—24no—
151H(5,𝔽2)—68no—
152H(5,𝔽2)—124no—
153C2×#41—28no—
154C2×#41—42no—
155C2×#41—64no—
156C2×#41—62no—
157C22×Q8D1211yescharacteristic, complemented
158C22×D4—34no—
159C22×D4—64no—
160C22×D4—62no—
161C22×D4D1211yescharacteristic, complemented
162C22×D4C22×D312yes—
163C23×C4D1212yescomplemented
164C23×C4C22×D311yescharacteristic
165C4⋊Q8—64no—
166#124—64no—
167#123—38no—
168#123—68no—
169#123—68no—
170#123—68no—
171#123—64no—
172#123—64no—
173#123—64no—
174#123—64no—
175C22⋊Q8—64no—
176C22⋊Q8—68no—
177#121—64no—
178#121—68no—
179#121—68no—
180#121—124no—
181#120—38no—
182#120—68no—
183#120—64no—
184#120—64no—
185C4×D4—68no—
186C4×D4—68no—
187#117—62no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—64no—
190C2×C22⋊C4—34no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—64no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4C22×D314yes—
195#99—28no—
196#99—616no—
197#95—42no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—81no—
201C23×C3—82no—
202C23×C3#12014yes—
203C23×C3C22×D411yescharacteristic
204C23×C3C22×D414yes—
205C23×C3H(5,𝔽2)12yes—
206C22×D3—24no—
207C22×D3—28no—
208C22×D3—44no—
209C22×D3—84no—
210C22×D3—84no—
211C3×Q8—44no—
212C3×D4—216no—
213C3×D4—44no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—48no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4—84no—
224C2×C3×C4#12014yes—
225#67—48no—
226#67—44no—
227#67—44no—
228#67—48no—
229#67—88no—
230C2×Dic3—28no—
231C2×Dic3—24no—
232C2×Dic3—28no—
233C2×Dic3—44no—
234C2×Dic3—84no—
235C2×Dic3—84no—
236D12—44no—
237D12—164no—
238C4×D3—48no—
239Dic6—44no—
240C24—34no—
241C24—62no—
242C24C2×D1212yes—
243C24C23×D311yescharacteristic
244#41—416no—
245#41—124no—
246#41—1216no—
247#41—124no—
248C2×Q8—24no—
249C2×Q8—41no—
250C2×Q8—62no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—41no—
255C2×D4—62no—
256C2×D4—64no—
257C2×D4—64no—
258C2×D4—68no—
259C2×D4—64no—
260C2×D4—68no—
261C2×D4—68no—
262C2×D4—124no—
263C2×D4—1216no—
264C2×D4—124no—
265C2×D4—124no—
266C2×D4D3×D418yes—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—34no—
273C22×C4—42no—
274C22×C4—64no—
275C22×C4—62no—
276C22×C4—64no—
277C22×C4—64no—
278C22×C4C2×D1211yescharacteristic
279C22×C4C2×D1211yescharacteristic
280C22×C4C2×D1212yes—
281C4⋊C4—68no—
282C4⋊C4—68no—
283C4⋊C4—64no—
284C4⋊C4—68no—
285C4⋊C4—124no—
286C22⋊C4—28no—
287C22⋊C4—38no—
288C22⋊C4—68no—
289C22⋊C4—68no—
290C22⋊C4—64no—
291C22⋊C4—68no—
292C22⋊C4—64no—
293C22⋊C4—68no—
294C22⋊C4—128no—
295C42—64no—
296C22×C3—24no—
297C22×C3—44no—
298C22×C3—42no—
299C22×C3—41no—
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—82no—
303C22×C3C2×#12012yes—
304C22×C3#53411yescharacteristic
305C22×C3D4214yes—
306C2×D3—48no—
307C2×D3—44no—
308C2×D3—88no—
309C2×D3—164no—
310C3×C4—24no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—84no—
314Dic3—48no—
315Dic3—44no—
316C23—22no—
317C23—22no—
318C23—64no—
319C23—68no—
320C23—81no—
321C23—82no—
322C23—124no—
323C23—244no—
324C23—244no—
325C23#88214yes—
326C23C22×D1211yescharacteristic
327C23C2×D3×D414yes—
328C23#100412yes—
329Q8—44no—
330Q8—124no—
331D4—216no—
332D4—44no—
333D4—124no—
334D4—128no—
335D4—124no—
336D4—124no—
337D4—128no—
338D4—248no—
339D4—484no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—68no—
350C2×C4—64no—
351C2×C4—68no—
352C2×C4—84no—
353C2×C4—128no—
354C2×C4—124no—
355C2×C4—244no—
356C2×C4—244no—
357C2×C4#88214yes—
358C2×C3—22no—
359C2×C3—24no—
360C2×C3—21no—
361C2×C3—44no—
362C2×C3#240311yescharacteristic
363D3—84no—
364C22—24no—
365C22—42no—
366C22—44no—
367C22—41no—
368C22—44no—
369C22—44no—
370C22—82no—
371C22—128no—
372C22—124no—
373C22—248no—
374C22—484no—
375C22C2×#88212yes—
376C22#549311yescharacteristic
377C22#559014yes—
378C4—24no—
379C4—41no—
380C4—41no—
381C4—84no—
382C4—128no—
383C4—124no—
384C3#2389611yescharacteristic, complemented, Sylow
385C2—24no—
386C2—22no—
387C2—21no—
388C2—44no—
389C2—244no—
390C2#7385211yescharacteristic
subgroupG/H
centerC2#73852
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1782C2
socleC2×C3#2403
hypercenter#1782D3
hypocenterC3#23896
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C24 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #87273C2semidirect
2. #78686C2semidirect
3. C3×#1782C2semidirect
4. #70024C2semidirect
5. #69539C2semidirect
8. C2×#1004C22semidirect
14. #5798C22semidirect
17. C2×#929C22semidirect
19. C2×C3×#99C22semidirect
20. C3×#381C22semidirect
21. C2×#806C22semidirect
24. #4526C22semidirect
29. #1782D3semidirect
32. C22×C3×Q8D4semidirect
33. C22×C3×D4D4semidirect
35. C23×C3×C4D4semidirect
54. #937D4semidirect
56. #935D4semidirect
92. C2×#99C2×D3semidirect
93. #381C2×D3semidirect
157. C22×Q8D12semidirect
161. C22×D4D12semidirect
163. C23×C4D12semidirect
384. C3#23896semidirect

Maximal subgroups

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

subgroupindex
#872732
#786862
C3×#17822
#700242
#695392
#238963