Finite Group Databasebeta

← #10690964

Subgroups — #10690964

Order 768.

692 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82699C211yesmaximal, characteristic
2C2×#4650C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic
4C2×#4537C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9156—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C22×Dic3⋊C4—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#4650—22no—
24#4650—22no—
25C3×#327—22no—
26#4550—22no—
27#4537—22no—
28#2854—31no—
29C2×#382—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—24no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—42no—
58C2×Dic3⋊C4—42no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3C2×C412yes—
64C2×C4×Dic3D412yes—
65C22×C3⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3×#95D412yes—
70C3⋊#95—24no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77C2×#123—61no—
78C2×#121C2×D311yescharacteristic
79C22×C4⋊C4—61no—
80C22×C42—31no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8C2×D311yescharacteristic
83#390—32no—
84#389—32no—
85#388—62no—
86#386—61no—
87#382—62no—
88#382—62no—
89#380—31no—
90C2×#95—31no—
91C2×#95—61no—
92C2×#95C2×D311yescharacteristic
93#336—62no—
94#327—22no—
95#327—62no—
96C24×C3—21no—
97C24×C3C2×D411yescharacteristic
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—42no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—42no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—42no—
129C22×Dic3—42no—
130C22×Dic3—42no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3C22⋊C412yes—
134C22×Dic3C22⋊C412yes—
135C22×Dic3D812yes—
136C22×Dic3SD1612yes—
137C2×C3×C8—42no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—44no—
144#217—41no—
145#217—41no—
146#217—44no—
147C4⋊Dic3—42no—
148Dic3⋊C4—48no—
149Dic3⋊C4—44no—
150C4×Dic3—22no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C4×Dic3—22no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C22×D4—21no—
157C22×D4C4×D311yescharacteristic
158C23×C4—21no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—61no—
162C23×C4C22×D311yescharacteristic
163C22×C8—21no—
164C22×C8—61no—
165#123—122no—
166#123—122no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—64no—
174C2×C4⋊C4—122no—
175C2×C4⋊C4—122no—
176C2×C4⋊C4C4×D311yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C42—32no—
182C2×C42—32no—
183C2×C42—61no—
184C2×C42—61no—
185C2×C42—61no—
186C2×C42—61no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191#95—21no—
192#95—32no—
193#95—62no—
194#95—64no—
195#95—61no—
196#95—61no—
197#95#6712yes—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3#12011yescharacteristic
207C23×C3#12311yescharacteristic
208C23×C3C2×D811yescharacteristic
209C23×C3C2×SD1611yescharacteristic
210C23×C3AGL(1,ℤ/8)11yescharacteristic
211C23×C3#13711yescharacteristic
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—82no—
231C2×C3×C4—82no—
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—82no—
254C2×Dic3—82no—
255C2×Dic3#10214yes—
256C3×C8—81no—
257C3⋊C8—81no—
258C24—21no—
259C24D3×D411yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—121no—
292C22×C4—121no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—122no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—122no—
306C4⋊C4—124no—
307C4⋊C4—128no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—62no—
315C42—62no—
316C42—62no—
317C42—62no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3#38611yescharacteristic
332C22×C3#41012yes—
333C22×C3C2×#10211yescharacteristic
334C22×C3#41711yescharacteristic
335C22×C3#42112yes—
336C22×C3#44711yescharacteristic
337C22×C3#44811yescharacteristic
338C22×C3#48011yescharacteristic
339C22×C3#48111yescharacteristic
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4—81no—
346Dic3—24no—
347Dic3—42no—
348Dic3—82no—
349Dic3—82no—
350C23—22no—
351C23—22no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23D3×C22⋊C411yescharacteristic
358C23#88311yescharacteristic
359C23D3×D811yescharacteristic
360C23#91111yescharacteristic
361C23D3×SD1611yescharacteristic
362C23#91511yescharacteristic
363C23#93711yescharacteristic
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—34no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—242no—
403C2×C4—242no—
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93811yescharacteristic
408C8—81no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—161no—
415C2×C3#151212yes—
416C2×C3#151712yes—
417C2×C3#187411yescharacteristic
418C2×C3#189711yescharacteristic
419C2×C3#189911yescharacteristic
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—81no—
429C22—81no—
430C22—82no—
431C22—82no—
432C22—81no—
433C22#474012yes—
434C22D3×#10211yescharacteristic
435C22#477111yescharacteristic
436C22#477411yescharacteristic
437C22#477711yescharacteristic
438C22#482112yes—
439C22#495611yescharacteristic
440C22#515611yescharacteristic
441C22#517011yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—64no—
447C4—82no—
448C4—81no—
449C4—122no—
450C4—242no—
451C4—242no—
452C3#915611yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—161no—
458C2#6828012yes—
459C2#6828212yes—
460C2#7002911yescharacteristic
461C2#7020011yescharacteristic
462C2#7038311yescharacteristic
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#9156
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×#4650C2semidirect
5. #69910C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
15. #5221C4semidirect
21. #4972C4semidirect
452. C3#9156semidirect

Maximal subgroups

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

subgroupindex
#826992
C2×#46502
C2×C3×#3272
C2×#45372
#699102
#676752
#676122
#91563