Finite Group Databasebeta

← #10695622

Subgroups — #10695622

Order 768.

812 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 99 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#84066C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic, complemented
3#78611C211yesmaximal, characteristic
4C3×#1787C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9948—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#947—21no—
11C22×#215—21no—
12C2×#930C2211yescharacteristic
13C2×#930C2211yescharacteristic
14C2×C3×#103—22no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5231—22no—
18#5229—21no—
19#5227—22no—
20#5227—22no—
21C2×#835—21no—
22#5213—22no—
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5115—22no—
26C2×#806—22no—
27#4736—21no—
28#4736—21no—
29#4732—21no—
30#4732—21no—
31#4451—22no—
32#3186—31no—
33C2×#448—31no—
34#2436—31no—
35#1964—31no—
36#1787D311yescharacteristic, complemented
37#1519—31no—
38#1282—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C22×#67—21no—
50C22×D12D411yescharacteristic
51C22×D12D411yescharacteristic
52C22×C3×C8D412yes—
53C2×C3×C4⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×#217D412yes—
56#947—44no—
57C2×#215—24no—
58C2×#215—44no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4×Dic3—21no—
73C22×C3⋊C8—21no—
74C2×D24—22no—
75C3×#103—44no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79#835—24no—
80C3⋊#95—24no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yes—
84#821—22no—
85#821—22no—
86#818—22no—
87#806—44no—
88C23×Q8C2×D311yescharacteristic
89C22×SD16—61no—
90C2×#124—61no—
91C2×#121—31no—
92C2×#121—31no—
93#480—61no—
94#480—61no—
95#466—62no—
96#448—62no—
97#448—62no—
98#447—61no—
99#447—61no—
100C2×#103—22no—
101C2×#103—61no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#393—62no—
106#391—61no—
107#386—62no—
108#380C2×D311yescharacteristic
109C4×C22⋊C4—31no—
110#327—62no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—44no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—44no—
124C2×C3×Q8—44no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C22×C3×C4—22no—
128C22×C3×C4—42no—
129C22×C3×C4—44no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—42no—
144C2×#67—42no—
145C2×#67—42no—
146C2×#67—42no—
147C22×Dic3—21no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12—42no—
155C2×D12—42no—
156C2×C3×C8—22no—
157C2×C3×C8—42no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—42no—
160#217—22no—
161#217—22no—
162#215—48no—
163(C2×D3)⋊C4—48no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4×Dic3—42no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—42no—
175D24—82no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—44no—
182C22×Q8D1212yes—
183C22×D4—31no—
184C22×D4—31no—
185C22×D4—61no—
186C22×D4—61no—
187C23×C4—21no—
188C23×C4C22×D311yescharacteristic
189C2×SD16—64no—
190C2×SD16—124no—
191C2×D8—62no—
192C22×C8—61no—
193C22×C8#6712yes—
194#124—124no—
195#121—61no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200#121—61no—
201#121—61no—
202#121—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C42—61no—
212#107—62no—
213#103—44no—
214#103—64no—
215#102—62no—
216#102—62no—
217#99—124no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—32no—
221C22⋊C8—61no—
222#95—21no—
223#95—64no—
224#95—61no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C23×C3#13512yes—
231C23×C3AGL(1,ℤ/8)12yes—
232C22×D3—42no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—42no—
237C22×D3—42no—
238C22×D3—81no—
239C22×D3—81no—
240C3×Q8—24no—
241C3×Q8—24no—
242C3×Q8—42no—
243C3×Q8—44no—
244C3×Q8—42no—
245C3×Q8—44no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—42no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264#67—82no—
265#67—82no—
266#67—81no—
267#67—81no—
268#67—81no—
269#67—81no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277D12—42no—
278D12—42no—
279D12—42no—
280D12—81no—
281C3×C8—42no—
282C3⋊C8—42no—
283C24—61no—
284C24—61no—
285C24D3×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—122no—
304C2×D4—122no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—122no—
308C2×D4—122no—
309C2×D4—122no—
310C2×D4—122no—
311C2×D4—121no—
312C2×D4—121no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—61no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4C2×D1211yescharacteristic
323C22×C4#23511yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4C2×#6711yescharacteristic
328SD16—128no—
329D8—242no—
330C2×C8—22no—
331C2×C8—42no—
332C2×C8—62no—
333C2×C8—122no—
334C4⋊C4—42no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—61no—
339C22⋊C4—42no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22⋊C4—128no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C42—122no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3#39211yescharacteristic
357C22×C3#44912yes—
358C22×C3#46312yes—
359C22×C3#49511yescharacteristic
360C22×C3#49611yescharacteristic
361C2×D3—84no—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—162no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—44no—
373C3×C4—161no—
374C3×C4B(4,𝔽2)12yes—
375Dic3—82no—
376Dic3—81no—
377Dic3—81no—
378C23—41no—
379C23—41no—
380C23—122no—
381C23—122no—
382C23—121no—
383C23—121no—
384C23—122no—
385C23—122no—
386C23—241no—
387C23—241no—
388C23#88211yescharacteristic
389C23#88411yescharacteristic
390C23#91412yes—
391C23#91912yes—
392C23#94011yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—42no—
396Q8—44no—
397Q8—42no—
398Q8—44no—
399D4—122no—
400D4—122no—
401D4—122no—
402D4—241no—
403D4—241no—
404D4—241no—
405D4—241no—
406D4—241no—
407D4—242no—
408D4—242no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4#88211yescharacteristic
431C2×C4#88411yescharacteristic
432C2×C4#93011yescharacteristic
433C2×C4#93711yescharacteristic
434C8—42no—
435C8—122no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3#200911yescharacteristic
441C2×C3#201211yescharacteristic
442C2×C3#202111yescharacteristic
443D3—161no—
444D3—161no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—244no—
454C22—241no—
455C22—241no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22—482no—
461C22#481012yes—
462C22#481912yes—
463C22#496011yescharacteristic
464C22#517611yescharacteristic
465C22#519111yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—161no—
471C4—242no—
472C4—241no—
473C4—241no—
474C4#503212yes—
475C3#994811yescharacteristic, complemented, Sylow
476C2—21no—
477C2—22no—
478C2—22no—
479C2—21no—
480C2—481no—
481C2—481no—
482C2#6950011yescharacteristic
483C2#7027711yescharacteristic
484C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC22×C3#392
prime radical#78611C2
hypercenter#1787D3
hypocenterC3#9948
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. #84066C2semidirect
2. C2×#5227C2semidirect
4. C3×#1787C2semidirect
5. #69528C2semidirect
6. #69509C2semidirect
7. #67677C2semidirect
36. #1787D3semidirect
475. C3#9948semidirect

Maximal subgroups

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

subgroupindex
#840662
C2×#52272
#786112
C3×#17872
#695282
#695092
#676772
#99483