Finite Group Databasebeta

← #10698045

Subgroups — #10698045

Order 768.

1066 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1066 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
1#84032C212yesmaximal, complemented
2#78611C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4#69944C212yesmaximal, complemented
5#67676C211yesmaximal, characteristic, complemented
6#10320—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×#953—22no—
9C2×#930—22no—
10C2×#930C2212yes—
11C2×C3×C22⋊C8—22no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5213—22no—
15#5213—24no—
16#5212—22no—
17#5212C2212yes—
18#5115—22no—
19#4736—22no—
20#4732—22no—
21#4470—24no—
22#3186—31no—
23#2847—32no—
24#1779D311yescharacteristic, complemented
25#1519—32no—
26#1289—31no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—21no—
30C23×C3×C4—22no—
31C23×C3×C4—21no—
32C23×C3×C4—44no—
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×#67—24no—
36C22×#67—22no—
37C22×D12D412yescomplemented
38C22×C3×C8—21no—
39C22×C3×C8D412yescomplemented
40#953—44no—
41#953—44no—
42C2×#217—24no—
43C2×#217—22no—
44#930—22no—
45#930—22no—
46#930—22no—
47#930—22no—
48#930—48no—
49C2×(C2×D3)⋊C4—24no—
50C2×(C2×D3)⋊C4—22no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3D412yescomplemented
53C2×D24—22no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—44no—
57C3⋊#95—24no—
58C3⋊#95D414yescomplemented
59#821—24no—
60#818—22no—
61C24×C4C2×D311yescharacteristic
62C2×#121—32no—
63C2×#121—62no—
64C2×#120—62no—
65#480—62no—
66#466—62no—
67#447—62no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2×D311yescharacteristic
70C2×C22⋊C8C2×D311yescharacteristic
71#386—64no—
72#386—62no—
73#380—32no—
74#380—62no—
75#327—64no—
76C24×C3—22no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—22no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3D812yescomplemented
83C24×C3C2×D411yescharacteristic
84C23×D3—22no—
85C22×C3×C4—24no—
86C22×C3×C4—28no—
87C22×C3×C4—22no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4—44no—
92C22×C3×C4—42no—
93C22×C3×C4—44no—
94C22×C3×C4—42no—
95C22×C3×C4—44no—
96C22×C3×C4—42no—
97C22×C3×C4—41no—
98C22×C3×C4—42no—
99C22×C3×C4—41no—
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—48no—
105C2×#67—48no—
106C2×#67—48no—
107C2×#67—42no—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—42no—
112C2×#67—42no—
113C22×Dic3—22no—
114C22×Dic3—24no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—44no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C2×C3×C8—42no—
121C2×C3×C8—41no—
122#217—48no—
123#217—42no—
124#217—42no—
125(C2×D3)⋊C4—48no—
126(C2×D3)⋊C4—42no—
127(C2×D3)⋊C4—42no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—44no—
131D24—82no—
132C25D1211yescharacteristic
133C22×D4—32no—
134C22×D4—64no—
135C22×D4—62no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—44no—
141C23×C4D1211yescharacteristic
142C23×C4C22×D311yescharacteristic
143C2×D8—62no—
144C22×C8—21no—
145C22×C8#6712yescomplemented
146#121—62no—
147#121—62no—
148#121—62no—
149#121—62no—
150#121—128no—
151#120—124no—
152#120—124no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—62no—
155C2×C22⋊C4—64no—
156C2×C22⋊C4—64no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—62no—
159#107—62no—
160#102—64no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—44no—
164#95—34no—
165#95—64no—
166C23×C3—24no—
167C23×C3—24no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—24no—
171C23×C3—22no—
172C23×C3—41no—
173C23×C3—44no—
174C23×C3—41no—
175C23×C3—42no—
176C23×C3—41no—
177C23×C3—42no—
178C23×C3—41no—
179C23×C3—48no—
180C23×C3—44no—
181C23×C3—41no—
182C23×C3—42no—
183C23×C3—41no—
184C23×C3#12012yes—
185C23×C3#12111yescharacteristic
186C23×C3C2×D811yescharacteristic
187C23×C3#13511yescharacteristic
188C23×C3AGL(1,ℤ/8)12yes—
189C22×D3—44no—
190C22×D3—44no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—24no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—44no—
201C2×C3×C4—48no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—44no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12112yes—
208C2×C3×C4#12711yescharacteristic
209#67—816no—
210#67—82no—
211#67—82no—
212#67—82no—
213#67—82no—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—42no—
217C2×Dic3—44no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220D12—44no—
221D12—82no—
222C3×C8—42no—
223C3×C8—81no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24—62no—
231C24D2412yescomplemented
232C24C2×D1211yescharacteristic
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—124no—
236C2×D4—128no—
237C2×D4—128no—
238C2×D4—128no—
239C2×D4—122no—
240C2×D4—122no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—122no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—64no—
261C22×C4—62no—
262C22×C4#23511yescharacteristic
263C22×C4D3×D412yes—
264C22×C4D3×D412yes—
265C22×C4C2×#6711yescharacteristic
266D8—242no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—124no—
274C22⋊C4—128no—
275C22⋊C4—128no—
276C22⋊C4—122no—
277C22⋊C4—122no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—28no—
285C22×C3—24no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3#39211yescharacteristic
299C22×C3#44712yes—
300C22×C3#44912yes—
301C22×C3#46611yescharacteristic
302C22×C3#46911yescharacteristic
303C2×D3—84no—
304C2×D3—84no—
305C2×D3—82no—
306C2×D3—82no—
307C2×D3—82no—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—42no—
311C3×C4B(4,𝔽2)12yes—
312Dic3—84no—
313Dic3—82no—
314C23—24no—
315C23—22no—
316C23—24no—
317C23—22no—
318C23—24no—
319C23—22no—
320C23—41no—
321C23—44no—
322C23—41no—
323C23—42no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—48no—
328C23—44no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—124no—
333C23—124no—
334C23—122no—
335C23—122no—
336C23—122no—
337C23#88212yes—
338C23C2×D2411yescharacteristic
339C23#90411yescharacteristic
340C23#90812yes—
341C23#93011yescharacteristic
342D4—124no—
343D4—242no—
344D4—2416no—
345D4—242no—
346D4—242no—
347D4—242no—
348D4—242no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—124no—
362C2×C4—124no—
363C2×C4—122no—
364C2×C4—124no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4#88412yes—
368C2×C4#93711yescharacteristic
369C2×C4#94011yescharacteristic
370C8—42no—
371C8—81no—
372C2×C3—22no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—44no—
377C2×C3#199911yescharacteristic
378C2×C3#201011yescharacteristic
379C2×C3#202111yescharacteristic
380D3—162no—
381C22—22no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—28no—
386C22—24no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—42no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—244no—
400C22—244no—
401C22—242no—
402C22—242no—
403C22—242no—
404C22#473212yes—
405C22#473412yes—
406C22#496011yescharacteristic
407C22#511511yescharacteristic
408C22#513511yescharacteristic
409C4—21no—
410C4—22no—
411C4—42no—
412C4—244no—
413C4—242no—
414C4#503212yes—
415C3#1032011yescharacteristic, complemented, Sylow
416C2—22no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—44no—
421C2—482no—
422C2#6950011yescharacteristic
423C2#6996111yescharacteristic
424C2#6999311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC22×C3#392
hypercenter#1779D3
hypocenterC3#10320
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. #84032C2semidirect
2. #78611C2semidirect
3. C3×#1779C2semidirect
4. #69944C2semidirect
5. #67676C2semidirect
24. #1779D3semidirect
37. C22×D12D4semidirect
39. C22×C3×C8D4semidirect
52. C2×C4⋊Dic3D4semidirect
58. C3⋊#95D4semidirect
82. C24×C3D8semidirect
145. C22×C8#67semidirect
231. C24D24semidirect
415. C3#10320semidirect

Maximal subgroups

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

subgroupindex
#840322
#786112
C3×#17792
#699442
#676762
#103203