Finite Group Databasebeta

← #10695719

Subgroups — #10695719

Order 768.

1160 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#5217C212yesmaximal, complemented
2#73381C211yesmaximal, characteristic, complemented
3C3×#1438C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9962—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4—22no—
9C22×#213—22no—
10C2×#930—22no—
11C2×#930C2212yes—
12C2×C3×#95C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5217—24no—
15#5217—24no—
16C2×#832—22no—
17C2×#830C2212yes—
18#4677—22no—
19#4671—22no—
20#4451—24no—
21C2×#447—32no—
22#2656—31no—
23C2×#327—32no—
24#1438D311yescharacteristic, complemented
25#1280—31no—
26C25×C3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—22no—
29C22×C3×D4—21no—
30C22×C3×D4—44no—
31C22×C3×D4D412yescomplemented
32C23×C3×C4—21no—
33C23×C3×C4D412yes—
34C23×C3×C4C2311yescharacteristic
35C22×#67—22no—
36C22×D12—22no—
37C22×D12D412yes—
38C2×C3×C22⋊C4—44no—
39C2×C3×C22⋊C4—44no—
40C2×C3×C22⋊C4—44no—
41C2×#213—28no—
42C2×#213—48no—
43#930—22no—
44#930—22no—
45#930—22no—
46#930—22no—
47#930—44no—
48#930—44no—
49C2×(C2×D3)⋊C4—24no—
50C2×(C2×D3)⋊C4—22no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3D412yes—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3×#95—22no—
56#832—28no—
57#830—22no—
58#830D414yescomplemented
59C23×D4C2×D311yescharacteristic
60C22×D8—62no—
61C2×#121—32no—
62C2×#121—62no—
63C22×C22⋊C4—22no—
64#447—64no—
65#447—64no—
66C2×#102—62no—
67C2×C22⋊C8—32no—
68#399—62no—
69#392—62no—
70C2×#95C2×D311yescharacteristic
71C2×#95C2×D311yescharacteristic
72#327—64no—
73C24×C3—24no—
74C24×C3—22no—
75C24×C3—42no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3—41no—
79C24×C3—41no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3C2×D411yescharacteristic
83C23×D3—22no—
84C2×C3×D4—28no—
85C2×C3×D4—24no—
86C2×C3×D4—44no—
87C2×C3×D4—44no—
88C2×C3×D4—48no—
89C2×C3×D4—44no—
90C2×C3×D4—42no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—42no—
101C22×C3×C4D814yescomplemented
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D412yes—
105C22×C3×C4#4111yescharacteristic
106C2×#67—42no—
107C2×#67—42no—
108C2×#67—44no—
109C2×#67—44no—
110C22×Dic3—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—44no—
114C2×D12—44no—
115C2×D12—82no—
116C2×D12—82no—
117C3×C22⋊C4—416no—
118#213—416no—
119(C2×D3)⋊C4—48no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—42no—
122C4⋊Dic3—22no—
123C4⋊Dic3—22no—
124C4⋊Dic3—44no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—44no—
127C25—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—32no—
132C22×D4—44no—
133C22×D4—62no—
134C22×D4—62no—
135C22×D4D1212yescomplemented
136C23×C4—21no—
137C23×C4#6712yes—
138C23×C4C22×D311yescharacteristic
139C2×D8—68no—
140C2×D8—128no—
141C22×C8—62no—
142#121—62no—
143#121—62no—
144#121—62no—
145#121—62no—
146#121—124no—
147#121—124no—
148C2×C4⋊C4—32no—
149C2×C4⋊C4—62no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—64no—
154C2×C22⋊C4—62no—
155#102—68no—
156C22⋊C8—34no—
157C22⋊C8—62no—
158#95—22no—
159#95—22no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—22no—
163C23×C3—24no—
164C23×C3—44no—
165C23×C3—44no—
166C23×C3—42no—
167C23×C3—44no—
168C23×C3—44no—
169C23×C3—44no—
170C23×C3—44no—
171C23×C3—42no—
172C23×C3—44no—
173C23×C3—44no—
174C23×C3—82no—
175C23×C3—82no—
176C23×C3—82no—
177C23×C3—82no—
178C23×C3—82no—
179C23×C3—82no—
180C23×C3—81no—
181C23×C3—81no—
182C23×C3—81no—
183C23×C3—81no—
184C23×C3#12011yescharacteristic
185C23×C3#12111yescharacteristic
186C23×C3#12711yescharacteristic
187C23×C3C2×D812yes—
188C23×C3AGL(1,ℤ/8)12yes—
189C22×D3—42no—
190C22×D3—44no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—44no—
194C3×D4—28no—
195C3×D4—44no—
196C3×D4—48no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4—82no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4#12012yes—
212C2×C3×C4#12112yes—
213#67—84no—
214#67—82no—
215#67—82no—
216C2×Dic3—42no—
217C2×Dic3—44no—
218D12—44no—
219D12—82no—
220D12—88no—
221C3⋊C8—44no—
222C24—24no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—62no—
232C24C2×#6711yescharacteristic
233C2×D4—28no—
234C2×D4—24no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—48no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—124no—
243C2×D4—124no—
244C2×D4—122no—
245C2×D4—124no—
246C2×D4—122no—
247C2×D4—124no—
248C2×D4—242no—
249C2×D4—242no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—62no—
261C22×C4#21314yescomplemented
262C22×C4C2×D1211yescharacteristic
263C22×C4#23511yescharacteristic
264C22×C4D3×D412yes—
265C22×C4D3×D412yes—
266D8—1216no—
267C2×C8—64no—
268C2×C8—124no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—124no—
272C22⋊C4—416no—
273C22⋊C4—128no—
274C22⋊C4—122no—
275C22⋊C4—122no—
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—48no—
285C22×C3—44no—
286C22×C3—84no—
287C22×C3—84no—
288C22×C3—84no—
289C22×C3—84no—
290C22×C3—84no—
291C22×C3—82no—
292C22×C3—82no—
293C22×C3#39211yescharacteristic
294C22×C3#44712yes—
295C22×C3#45311yescharacteristic
296C22×C3#45411yescharacteristic
297C22×C3B(4,𝔽2)12yes—
298C22×C3#45912yes—
299C22×C3#49311yescharacteristic
300C22×C3#49611yescharacteristic
301C2×D3—84no—
302C2×D3—84no—
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—82no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—44no—
309C3×C4—82no—
310Dic3—82no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23—24no—
315C23—44no—
316C23—44no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—122no—
336C23—124no—
337C23—124no—
338C23—122no—
339C23—122no—
340C23#93011yescharacteristic
341C23C2×#21312yes—
342C23#93711yescharacteristic
343C23#94011yescharacteristic
344C23#94912yes—
345D4—28no—
346D4—44no—
347D4—48no—
348D4—124no—
349D4—242no—
350D4—248no—
351D4—244no—
352D4—242no—
353D4—242no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—122no—
369C2×C4—124no—
370C2×C4#88212yes—
371C2×C4#88412yes—
372C8—124no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—84no—
377C2×C3#159512yes—
378C2×C3#159812yes—
379C2×C3#200611yescharacteristic
380C2×C3#201111yescharacteristic
381C2×C3#202111yescharacteristic
382D3—162no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—48no—
392C22—44no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—82no—
397C22—82no—
398C22—84no—
399C22—84no—
400C22—242no—
401C22—242no—
402C22—244no—
403C22—244no—
404C22—242no—
405C22#496011yescharacteristic
406C22#501512yes—
407C22#503212yes—
408C22#503612yes—
409C22#505811yescharacteristic
410C22#506211yescharacteristic
411C22#507311yescharacteristic
412C22#507711yescharacteristic
413C4—22no—
414C4—22no—
415C4—44no—
416C4—82no—
417C4—242no—
418C3#996211yescharacteristic, complemented, Sylow
419C2—21no—
420C2—22no—
421C2—21no—
422C2—84no—
423C2—482no—
424C2#6950011yescharacteristic
425C2#6965811yescharacteristic
426C2#6966411yescharacteristic
427C2#6969612yes—
428C2#6970312yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1438C2
socleC23×C3#120
hypercenter#1438D3
hypocenterC3#9962
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
1. C2×#5217C2semidirect
2. #73381C2semidirect
3. C3×#1438C2semidirect
4. C2×#4451C2semidirect
5. #67613C2semidirect
24. #1438D3semidirect
31. C22×C3×D4D4semidirect
58. #830D4semidirect
101. C22×C3×C4D8semidirect
135. C22×D4D12semidirect
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
C2×#52172
#733812
C3×#14382
C2×#44512
#676132
#99623