Finite Group Databasebeta

← #10695619

Subgroups — #10695619

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×#3186C211yesmaximal, characteristic
2#82654C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#69963C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9948—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#936—21no—
13C22×#214—21no—
14C3×#480—21no—
15C3×#480—21no—
16C3×#466—22no—
17C3×#447—21no—
18C3×#447—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5220—22no—
21C2×#834—22no—
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#816—22no—
26#5040—22no—
27#5040—22no—
28#4973—21no—
29C2×#810—21no—
30#4947—22no—
31#4537—22no—
32#3186D311yescharacteristic
33C2×#448—31no—
34#2436—31no—
35#1964—31no—
36#1787—31no—
37#1519—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—44no—
51C22×Dic6D412yes—
52C2×C3×D8—22no—
53C22×C3×C8D412yes—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—24no—
67C2×#217—21no—
68C2×#217D412yes—
69#936—44no—
70C2×#214—24no—
71C2×#214—44no—
72C2×C4⋊Dic3—21no—
73C2×C4×Dic3—21no—
74C22×C3⋊C8—21no—
75C3×#107—22no—
76C3×#102—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80#834—44no—
81C3⋊#95—24no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yes—
86#816—44no—
87#810—24no—
88C23×Q8—31no—
89C22×SD16—61no—
90C2×#124—61no—
91C2×#121C2×D311yescharacteristic
92C2×#121C2×D311yescharacteristic
93#480—21no—
94#480—21no—
95#466—22no—
96#448—62no—
97#448—62no—
98#447—21no—
99#447—21no—
100C2×#103—61no—
101C2×#103—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#393—62no—
106#391—61no—
107#386—62no—
108#380—31no—
109C4×C22⋊C4—31no—
110#327—62no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3C2×D411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—44no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C2×Dic6—24no—
144C2×Dic6—24no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—24no—
148C2×Dic6—44no—
149C2×Dic6—44no—
150C2×Dic6—44no—
151C2×Dic6—42no—
152C2×Dic6—42no—
153C2×Dic6—42no—
154C2×Dic6—41no—
155C2×Dic6—41no—
156C3×D8—82no—
157C2×C3×C8—22no—
158C2×C3×C8—42no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167#217—22no—
168#217—22no—
169#217—48no—
170#217—42no—
171#214—48no—
172C4⋊Dic3—42no—
173C4×Dic3—42no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—42no—
176C22×Q8—32no—
177C22×Q8—61no—
178C22×Q8—61no—
179C22×Q8—61no—
180C22×Q8—61no—
181C22×Q8—61no—
182C22×Q8—124no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4#6711yescharacteristic
186C22×D4#6711yescharacteristic
187C23×C4—61no—
188C23×C4C22×D311yescharacteristic
189C2×SD16—64no—
190C2×SD16—124no—
191C2×D8—22no—
192C22×C8—61no—
193C22×C8#6712yes—
194#124—124no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202#121—21no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—32no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—61no—
211C2×C42—61no—
212#107—22no—
213#103—64no—
214#103—124no—
215#102—22no—
216#102—22no—
217#99—124no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—32no—
221C22⋊C8—61no—
222#95—64no—
223#95—61no—
224#95—61no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—81no—
234C23×C3—81no—
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3#12711yescharacteristic
238C23×C3#13512yes—
239C23×C3AGL(1,ℤ/8)12yes—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—81no—
244C3×D4—82no—
245C3×D4—82no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—81no—
249C3×D4—81no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×Dic3—24no—
265C2×Dic3—44no—
266C2×Dic3—44no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275Dic6—24no—
276Dic6—24no—
277Dic6—42no—
278Dic6—42no—
279Dic6—44no—
280Dic6—44no—
281C3×C8—42no—
282C3⋊C8—42no—
283C24—21no—
284C24—21no—
285C24D3×D411yescharacteristic
286C2×Q8—64no—
287C2×Q8—64no—
288C2×Q8—64no—
289C2×Q8—62no—
290C2×Q8—62no—
291C2×Q8—122no—
292C2×Q8—122no—
293C2×Q8—122no—
294C2×Q8—124no—
295C2×Q8—124no—
296C2×Q8—124no—
297C2×Q8—121no—
298C2×Q8—121no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—124no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4#23711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328SD16—128no—
329D8—82no—
330C2×C8—22no—
331C2×C8—42no—
332C2×C8—62no—
333C2×C8—122no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—122no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—62no—
344C22⋊C4—62no—
345C22⋊C4—128no—
346C22⋊C4—122no—
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—84no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—162no—
364C22×C3#39211yescharacteristic
365C22×C3#44912yes—
366C22×C3#46312yes—
367C22×C3#49511yescharacteristic
368C22×C3#49611yescharacteristic
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4B(4,𝔽2)12yes—
375Dic3—44no—
376Dic3—82no—
377Dic3—161no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—81no—
387C23—81no—
388C23#91112yes—
389C23#91212yes—
390C23#93711yescharacteristic
391C23#93911yescharacteristic
392C23#94011yescharacteristic
393Q8—64no—
394Q8—64no—
395Q8—122no—
396Q8—122no—
397Q8—124no—
398Q8—124no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—81no—
403D4—82no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—64no—
420C2×C4—122no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4—241no—
429C2×C4—241no—
430C2×C4#93711yescharacteristic
431C2×C4#93811yescharacteristic
432C2×C4#93911yescharacteristic
433C2×C4#95311yescharacteristic
434C8—42no—
435C8—122no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—161no—
441C2×C3—161no—
442C2×C3#200911yescharacteristic
443C2×C3#201211yescharacteristic
444C2×C3#202111yescharacteristic
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—81no—
458C22—81no—
459C22—81no—
460C22—162no—
461C22#515312yes—
462C22#515411yescharacteristic
463C22#515511yescharacteristic
464C22#515812yes—
465C22#522311yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—81no—
470C4—81no—
471C4—124no—
472C4—242no—
473C4—481no—
474C4#524412yes—
475C3#994811yescharacteristic, complemented, Sylow
476C2—22no—
477C2—21no—
478C2—22no—
479C2—21no—
480C2—161no—
481C2—161no—
482C2#7185111yescharacteristic
483C2#7204511yescharacteristic
484C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3186C2
socleC22×C3#392
hypercenter#3186D3
hypocenterC3#9948
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
2. #82654C2semidirect
3. C2×#5040C2semidirect
4. #71686C2semidirect
5. #71675C2semidirect
6. #69963C2semidirect
7. #67677C2semidirect
475. C3#9948semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#826542
C2×#50402
#716862
#716752
#699632
#676772
#99483