Finite Group Databasebeta

← #10690929

Subgroups — #10690929

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82699C211yesmaximal, characteristic
2C2×#4648C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic
4C2×#4537C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9149—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C22×C4×Dic3—21no—
12C22×C4×Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5221C412yescomplemented
16#5220—21no—
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#4972C412yescomplemented
22#4944—22no—
23#4648—22no—
24#4648—22no—
25C3×#327—22no—
26#4550—22no—
27#4537—22no—
28#2854—31no—
29C2×#376—31no—
30#1817—31no—
31C2×#327—31no—
32C2×#327D311yescharacteristic
33#1294—31no—
34#1294—31no—
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—21no—
40C23×Dic3D411yescharacteristic
41C23×Dic3D411yescharacteristic
42C22×C3×C8—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—22no—
51C2×#217—22no—
52#935—42no—
53#935—42no—
54C2×C4⋊Dic3C2×C412yes—
55C2×Dic3⋊C4—22no—
56C2×C4×Dic3—24no—
57C2×C4×Dic3—28no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3C2×C412yes—
63C2×C4×Dic3D412yes—
64C22×C3⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—24no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×#123—61no—
77C2×#121C2×D311yescharacteristic
78C22×C42—31no—
79C22×C42—61no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#390—32no—
83#389—32no—
84#388—62no—
85#386—61no—
86#380—31no—
87#376—62no—
88#376—62no—
89C2×#95—31no—
90C2×#95—61no—
91C2×#95C2×D311yescharacteristic
92#336—62no—
93#327—22no—
94#327—62no—
95C24×C3—21no—
96C24×C3C2×D411yescharacteristic
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—42no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—42no—
110C22×C3×C4—42no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—24no—
122C22×Dic3—24no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—42no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C22×Dic3C22⋊C412yes—
135C22×Dic3C22⋊C412yes—
136C2×C3×C8—42no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142#217—44no—
143#217—41no—
144#217—41no—
145#217—44no—
146C4⋊Dic3—42no—
147Dic3⋊C4—44no—
148C4×Dic3—216no—
149C4×Dic3—22no—
150C4×Dic3—22no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—41no—
155C22×D4—21no—
156C22×D4C4×D311yescharacteristic
157C23×C4—21no—
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—61no—
161C23×C4C22×D311yescharacteristic
162C22×C8—21no—
163C22×C8—61no—
164#123—122no—
165#123—122no—
166#121—21no—
167#121—21no—
168#121—21no—
169#121—21no—
170C2×C4⋊C4—32no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4C4×D311yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—61no—
177C2×C42—32no—
178C2×C42—32no—
179C2×C42—64no—
180C2×C42—68no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42—61no—
184C2×C42—61no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#95—21no—
190#95—32no—
191#95—62no—
192#95—64no—
193#95—61no—
194#95—61no—
195#95#6712yes—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3#12011yescharacteristic
205C23×C3#12311yescharacteristic
206C23×C3#13511yescharacteristic
207C23×C3#13511yescharacteristic
208C23×C3AGL(1,ℤ/8)11yescharacteristic
209C23×C3AGL(1,ℤ/8)11yescharacteristic
210C3×D4—81no—
211C3×D4—81no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—24no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4—44no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—24no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—42no—
243C2×Dic3—44no—
244C2×Dic3—44no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3#10414yes—
254C3×C8—81no—
255C3⋊C8—81no—
256C24—21no—
257C24D3×D411yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—64no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—64no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—122no—
289C22×C4—121no—
290C22×C4—121no—
291C22×C4C2×C4×D311yescharacteristic
292C22×C4#23511yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4#23711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—122no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—122no—
304C4⋊C4—124no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—124no—
308C22⋊C4—124no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C42—616no—
312C42—62no—
313C42—62no—
314C42—62no—
315C42—62no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—82no—
328C22×C3—82no—
329C22×C3#38611yescharacteristic
330C22×C3#41012yes—
331C22×C3#41611yescharacteristic
332C22×C3#41811yescharacteristic
333C22×C3C2×#10412yes—
334C22×C3#44911yescharacteristic
335C22×C3#44911yescharacteristic
336C22×C3#48211yescharacteristic
337C22×C3#48211yescharacteristic
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—81no—
344Dic3—24no—
345Dic3—44no—
346Dic3—42no—
347Dic3—82no—
348C23—22no—
349C23—22no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23D3×C22⋊C411yescharacteristic
356C23#88311yescharacteristic
357C23#91111yescharacteristic
358C23#91211yescharacteristic
359C23#91411yescharacteristic
360C23#91611yescharacteristic
361C23#93711yescharacteristic
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—34no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—64no—
384C2×C4—64no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—64no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—122no—
392C2×C4—124no—
393C2×C4—124no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4#67⋊C411yescharacteristic
403C2×C4#88411yescharacteristic
404C2×C4C4×#6711yescharacteristic
405C2×C4#93811yescharacteristic
406C8—81no—
407C8—241no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—161no—
413C2×C3#150012yes—
414C2×C3#150512yes—
415C2×C3#187511yescharacteristic
416C2×C3#189611yescharacteristic
417C2×C3#189911yescharacteristic
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—82no—
429C22—82no—
430C22—81no—
431C22#474012yes—
432C22#477011yescharacteristic
433C22#477211yescharacteristic
434C22#477911yescharacteristic
435C22#478411yescharacteristic
436C22D3×#10412yes—
437C22#495611yescharacteristic
438C22#515811yescharacteristic
439C22#517211yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—64no—
446C4—81no—
447C4—124no—
448C4—122no—
449C4—242no—
450C3#914911yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—161no—
456C2#6827612yes—
457C2#6827812yes—
458C2#7002911yescharacteristic
459C2#7020111yescharacteristic
460C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9149
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#4648C2semidirect
5. #69910C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
15. #5221C4semidirect
21. #4972C4semidirect
450. C3#9149semidirect

Maximal subgroups

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

subgroupindex
#826992
C2×#46482
C2×C3×#3272
C2×#45372
#699102
#676752
#676122
#91493