Finite Group Databasebeta

← #10695621

Subgroups — #10695621

Order 768.

812 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2436C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic, complemented
3#82625C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9948—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×#121C2211yescharacteristic
12C22×#215—21no—
13C2×#930C2211yescharacteristic
14C3×#393—22no—
15C3×#391—21no—
16C3×#386—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5227—22no—
19#5227—22no—
20C2×#835—21no—
21C2×#835—22no—
22#5217—21no—
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5038—22no—
27#5036—21no—
28#5033—21no—
29#4967—21no—
30C2×#806—22no—
31#4537—22no—
32#3186—31no—
33C2×#448—31no—
34#2436D311yescharacteristic, complemented
35#1964—31no—
36#1787—31no—
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—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C22×#67—21no—
51C22×D12D411yescharacteristic
52C3×#124—44no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4D412yes—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×#215—24no—
64C2×#215—44no—
65C2×#213—22no—
66#930—21no—
67#930—21no—
68#930—21no—
69#930—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C22×C3⋊C8—21no—
74C22×C3⋊C8D412yes—
75C3×#95—24no—
76C3×#95—21no—
77#835—24no—
78#835—44no—
79#832—22no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#809—22no—
86#807—22no—
87#806—44no—
88C23×Q8C2×D311yescharacteristic
89C22×SD16—61no—
90C2×#124—21no—
91C2×#121—31no—
92C2×#121C2×D311yescharacteristic
93#480—61no—
94#480—61no—
95#466—62no—
96#448—62no—
97#448—62no—
98#447—61no—
99#447—61no—
100C2×#103—61no—
101C2×#103—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—31no—
105#393—22no—
106#391—21no—
107#386—22no—
108#380—31no—
109C4×C22⋊C4C2×D311yescharacteristic
110#327—62no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
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—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—42no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—42no—
133C2×C3×D4—42no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—42no—
138C22×C3×C4—44no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—42no—
150C2×#67—42no—
151C22×Dic3—21no—
152C22×Dic3—41no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—42no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—48no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C42—42no—
164#217—42no—
165#215—48no—
166#213—82no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—42no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—44no—
182C22×Q8#6712yes—
183C22×D4—21no—
184C22×D4—31no—
185C22×D4—61no—
186C22×D4#6711yescharacteristic
187C23×C4—21no—
188C23×C4C22×D311yescharacteristic
189C2×SD16—64no—
190C2×SD16—124no—
191C2×D8—62no—
192C22×C8—32no—
193C22×C8—61no—
194#124—44no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—61no—
200#121—61no—
201#121—61no—
202#121—61no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—24no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4#6712yes—
211C2×C42—21no—
212#107—62no—
213#103—64no—
214#103—124no—
215#102—62no—
216#102—62no—
217#99—124no—
218C22⋊C8—32no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222#95—24no—
223#95—21no—
224#95—61no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3#12011yescharacteristic
232C23×C3#12111yescharacteristic
233C23×C3#12711yescharacteristic
234C23×C3#13512yes—
235C23×C3AGL(1,ℤ/8)12yes—
236C22×D3—41no—
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—
246C3×D4—42no—
247C3×D4—81no—
248C3×D4—82no—
249C3×D4—81no—
250C3×D4—81no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271#67—82no—
272#67—81no—
273#67—81no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279D12—42no—
280D12—42no—
281C3⋊C8—42no—
282C3⋊C8—42no—
283C24—21no—
284C24—61no—
285C24C2×#6711yescharacteristic
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—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—122no—
309C2×D4—121no—
310C2×D4—122no—
311C2×D4—122no—
312C2×D4—121no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—61no—
321C22×C4—121no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4#23711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328SD16—128no—
329D8—242no—
330C2×C8—62no—
331C2×C8—62no—
332C2×C8—122no—
333C2×C8—122no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—122no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—48no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—122no—
347C42—42no—
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—84no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3#39211yescharacteristic
361C22×C3#44912yes—
362C22×C3#46312yes—
363C22×C3#49511yescharacteristic
364C22×C3#49611yescharacteristic
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—82no—
374C3×C4—81no—
375C3×C4B(4,𝔽2)12yes—
376Dic3—81no—
377Dic3—161no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—121no—
385C23—122no—
386C23—241no—
387C23—241no—
388C23#93811yescharacteristic
389C23#94011yescharacteristic
390C23#94912yes—
391C23#95012yes—
392C23#95311yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—42no—
396Q8—44no—
397Q8—42no—
398Q8—44no—
399D4—42no—
400D4—81no—
401D4—82no—
402D4—81no—
403D4—81no—
404D4—122no—
405D4—122no—
406D4—242no—
407D4—241no—
408D4—241no—
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—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—121no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4—241no—
429C2×C4—241no—
430C2×C4#93711yescharacteristic
431C2×C4#93711yescharacteristic
432C2×C4#93911yescharacteristic
433C2×C4#93911yescharacteristic
434C8—122no—
435C8—122no—
436C2×C3—21no—
437C2×C3—22no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—161no—
441C2×C3#200911yescharacteristic
442C2×C3#201211yescharacteristic
443C2×C3#202111yescharacteristic
444D3—161no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—84no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22—482no—
461C22#505712yes—
462C22#505811yescharacteristic
463C22#505911yescharacteristic
464C22#522311yescharacteristic
465C22#523812yes—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—82no—
471C4—81no—
472C4—241no—
473C4—481no—
474C4#503212yes—
475C3#994811yescharacteristic, complemented, Sylow
476C2—22no—
477C2—21no—
478C2—22no—
479C2—21no—
480C2—161no—
481C2—481no—
482C2#7167111yescharacteristic
483C2#7172811yescharacteristic
484C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2436C2
socleC22×C3#392
prime radical#82625C2
hypercenter#2436D3
hypocenterC3#9948
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. C3×#2436C2semidirect
2. C2×#5227C2semidirect
4. #72132C2semidirect
5. #71675C2semidirect
6. #69456C2semidirect
7. #67617C2semidirect
34. #2436D3semidirect
475. C3#9948semidirect

Maximal subgroups

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

subgroupindex
C3×#24362
C2×#52272
#826252
#721322
#716752
#694562
#676172
#99483