Finite Group Databasebeta

← #10695717

Subgroups — #10695717

Order 768.

1160 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#2656C211yesmaximal, characteristic, complemented
2C2×#5036C212yesmaximal, complemented
3C2×#4537C212yesmaximal, complemented
4#69426C211yesmaximal, characteristic, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9962—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×#121—22no—
9C2×C3×#121C2212yes—
10C22×#213—22no—
11C22×(C2×D3)⋊C4—22no—
12C3×#399—22no—
13C3×#392—22no—
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5036—24no—
18#5036—24no—
19C2×#809—22no—
20#4537—24no—
21C2×#447—32no—
22#2656D311yescharacteristic, complemented
23C2×#327—32no—
24#1438—31no—
25#1280—31no—
26C24×D3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—22no—
29C22×C3×D4D412yes—
30C23×C3×C4D412yes—
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—21no—
33C22×D12—22no—
34C22×D12—22no—
35C22×D12—21no—
36C22×D12—44no—
37C22×D12D412yescomplemented
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—22no—
42C3×#121—44no—
43C3×#121—44no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4—22no—
48C2×#213—28no—
49C2×#213—48no—
50C2×(C2×D3)⋊C4—44no—
51C2×(C2×D3)⋊C4—44no—
52C2×(C2×D3)⋊C4—44no—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3⋊#95—22no—
56#830—22no—
57#830D414yescomplemented
58#809—28no—
59C23×D4—31no—
60C22×D8—62no—
61C2×#121—22no—
62C2×#121C2×D312yes—
63C22×C22⋊C4—62no—
64#447—64no—
65#447—64no—
66C2×#102—62no—
67C2×C22⋊C8—32no—
68#399—22no—
69#392—22no—
70C2×#95—31no—
71C2×#95C2×D311yescharacteristic
72#327—64no—
73C24×C3—22no—
74C24×C3C2×D411yescharacteristic
75C23×D3—24no—
76C23×D3—22no—
77C23×D3—42no—
78C23×D3—42no—
79C23×D3—42no—
80C23×D3—41no—
81C23×D3—41no—
82C23×D3—42no—
83C23×D3—42no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—44no—
87C2×C3×D4—44no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—44no—
91C2×C3×D4—44no—
92C2×C3×D4—82no—
93C2×C3×D4—82no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4D814yescomplemented
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D412yes—
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—22no—
106C22×Dic3—42no—
107C22×Dic3—41no—
108C22×Dic3—41no—
109C22×Dic3—42no—
110C2×D12—28no—
111C2×D12—24no—
112C2×D12—44no—
113C2×D12—48no—
114C2×D12—44no—
115C2×D12—44no—
116C2×D12—42no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—44no—
120C3×C22⋊C4—48no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—42no—
123#213—416no—
124(C2×D3)⋊C4—416no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—44no—
127C25—61no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—32no—
131C22×D4—62no—
132C22×D4—61no—
133C22×D4—62no—
134C22×D4—124no—
135C22×D4#6712yes—
136C23×C4—61no—
137C23×C4#6712yes—
138C23×C4C22×D311yescharacteristic
139C2×D8—68no—
140C2×D8—128no—
141C22×C8—62no—
142#121—22no—
143#121—22no—
144#121—22no—
145#121—22no—
146#121—44no—
147#121—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4#6712yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—124no—
153C2×C22⋊C4—124no—
154C2×C22⋊C4—124no—
155#102—68no—
156C22⋊C8—34no—
157C22⋊C8—62no—
158#95—22no—
159#95—62no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—22no—
163C23×C3—42no—
164C23×C3—44no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—44no—
168C23×C3#12011yescharacteristic
169C23×C3#12111yescharacteristic
170C23×C3#12711yescharacteristic
171C23×C3C2×D812yes—
172C23×C3AGL(1,ℤ/8)12yes—
173C22×D3—24no—
174C22×D3—44no—
175C22×D3—44no—
176C22×D3—42no—
177C22×D3—44no—
178C22×D3—44no—
179C22×D3—44no—
180C22×D3—44no—
181C22×D3—42no—
182C22×D3—44no—
183C22×D3—44no—
184C22×D3—82no—
185C22×D3—82no—
186C22×D3—82no—
187C22×D3—82no—
188C22×D3—82no—
189C22×D3—82no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C3×D4—44no—
195C3×D4—82no—
196C3×D4—88no—
197C3×D4—84no—
198C3×D4—82no—
199C3×D4—82no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4—82no—
211C2×C3×C4#12012yes—
212C2×C3×C4#12112yes—
213C2×Dic3—44no—
214C2×Dic3—42no—
215C2×Dic3—82no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218D12—28no—
219D12—44no—
220D12—48no—
221C3⋊C8—44no—
222C24—22no—
223C24—64no—
224C24—62no—
225C24—122no—
226C24—122no—
227C24—122no—
228C24—122no—
229C24—122no—
230C24—121no—
231C24—121no—
232C24C2×#6711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—68no—
242C2×D4—64no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—124no—
246C2×D4—124no—
247C2×D4—128no—
248C2×D4—124no—
249C2×D4—122no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—62no—
257C22×C4—122no—
258C22×C4—121no—
259C22×C4—122no—
260C22×C4—121no—
261C22×C4#21314yescomplemented
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D412yes—
264C22×C4#23711yescharacteristic
265C22×C4C2×#6712yes—
266D8—1216no—
267C2×C8—64no—
268C2×C8—124no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C22⋊C4—48no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—1216no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—84no—
285C22×C3—84no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3#39211yescharacteristic
290C22×C3#44712yes—
291C22×C3#45311yescharacteristic
292C22×C3#45411yescharacteristic
293C22×C3B(4,𝔽2)12yes—
294C22×C3#45912yes—
295C22×C3#49311yescharacteristic
296C22×C3#49611yescharacteristic
297C2×D3—48no—
298C2×D3—44no—
299C2×D3—84no—
300C2×D3—84no—
301C2×D3—84no—
302C2×D3—84no—
303C2×D3—84no—
304C2×D3—82no—
305C2×D3—82no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—44no—
309C3×C4—82no—
310Dic3—82no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—64no—
320C23—124no—
321C23—124no—
322C23—122no—
323C23—124no—
324C23—124no—
325C23—124no—
326C23—124no—
327C23—122no—
328C23—124no—
329C23—124no—
330C23—242no—
331C23—242no—
332C23—242no—
333C23—242no—
334C23—242no—
335C23—242no—
336C23—241no—
337C23—241no—
338C23—241no—
339C23—241no—
340C23C2×#21312yes—
341C23#93212yes—
342C23#93811yescharacteristic
343C23#94011yescharacteristic
344C23#95311yescharacteristic
345D4—44no—
346D4—68no—
347D4—82no—
348D4—88no—
349D4—84no—
350D4—82no—
351D4—82no—
352D4—124no—
353D4—128no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—124no—
366C2×C4—122no—
367C2×C4—241no—
368C2×C4—242no—
369C2×C4—241no—
370C2×C4#93712yes—
371C2×C4#93912yes—
372C8—124no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—162no—
377C2×C3#159512yes—
378C2×C3#159812yes—
379C2×C3#200611yescharacteristic
380C2×C3#201111yescharacteristic
381C2×C3#202111yescharacteristic
382D3—84no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—84no—
393C22—82no—
394C22—84no—
395C22—82no—
396C22—128no—
397C22—124no—
398C22—244no—
399C22—244no—
400C22—244no—
401C22—242no—
402C22—244no—
403C22—244no—
404C22—242no—
405C22#507212yes—
406C22#507311yescharacteristic
407C22#507411yescharacteristic
408C22#521712yes—
409C22#522311yescharacteristic
410C22#524111yescharacteristic
411C22#524412yes—
412C22#524511yescharacteristic
413C4—22no—
414C4—22no—
415C4—44no—
416C4—82no—
417C4—242no—
418C3#996211yescharacteristic, complemented, Sylow
419C2—21no—
420C2—22no—
421C2—21no—
422C2—162no—
423C2—244no—
424C2#7176711yescharacteristic
425C2#7184512yes—
426C2#7185111yescharacteristic
427C2#7187812yes—
428C2#7187911yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2656C2
socleC23×C3#120
hypercenter#2656D3
hypocenterC3#9962
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C3×#2656C2semidirect
2. C2×#5036C2semidirect
3. C2×#4537C2semidirect
4. #69426C2semidirect
5. #67613C2semidirect
22. #2656D3semidirect
37. C22×D12D4semidirect
57. #830D4semidirect
100. C22×C3×C4D8semidirect
261. C22×C4#213semidirect
418. C3#9962semidirect

Maximal subgroups

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

subgroupindex
C3×#26562
C2×#50362
C2×#45372
#694262
#676132
#99623