Finite Group Databasebeta

← #10690969

Subgroups — #10690969

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82699C211yesmaximal, characteristic
2C2×#4651C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic
4C2×#4537C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9157—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C22×C4×Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5221C412yescomplemented
15#5220—21no—
16#5212C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#4972C412yescomplemented
22#4944—22no—
23#4651—22no—
24#4651—22no—
25C3×#327—22no—
26#4550—22no—
27#4537—22no—
28#2854—31no—
29C2×#383—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—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3C2×C412yes—
61C2×C4×Dic3D412yes—
62C22×C3⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66C3×#95D412yes—
67C3⋊#95—24no—
68C3⋊#95—24no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×#123—61no—
76C2×#121C2×D311yescharacteristic
77C22×C42—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8C2×D311yescharacteristic
80#390—32no—
81#389—32no—
82#388—62no—
83#386—61no—
84#383—62no—
85#383—62no—
86#380—31no—
87C2×#95—31no—
88C2×#95—61no—
89C2×#95—61no—
90C2×#95C2×D311yescharacteristic
91#336—62no—
92#327—22no—
93#327—62no—
94C24×C3—21no—
95C24×C3C2×D411yescharacteristic
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—42no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
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—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141#217—44no—
142#217—44no—
143#217—41no—
144#217—41no—
145C4⋊Dic3—42no—
146Dic3⋊C4—44no—
147C4×Dic3—22no—
148C4×Dic3—22no—
149C4×Dic3—22no—
150C4×Dic3—22no—
151C2×C3⋊C8—42no—
152C2×C3⋊C8—41no—
153C22×D4—21no—
154C22×D4C4×D311yescharacteristic
155C23×C4—21no—
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—61no—
159C23×C4C22×D311yescharacteristic
160C22×C8—21no—
161C22×C8—61no—
162#123—122no—
163#123—122no—
164#121—21no—
165#121—21no—
166#121—21no—
167#121—21no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4C4×D311yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—61no—
175C2×C42—32no—
176C2×C42—32no—
177C2×C42—61no—
178C2×C42—61no—
179C2×C42—61no—
180C2×C42—61no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185#95—21no—
186#95—32no—
187#95—62no—
188#95—64no—
189#95—64no—
190#95—61no—
191#95—61no—
192#95#6712yes—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12311yescharacteristic
203C23×C3#13511yescharacteristic
204C23×C3#13511yescharacteristic
205C23×C3AGL(1,ℤ/8)11yescharacteristic
206C23×C3AGL(1,ℤ/8)11yescharacteristic
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C3×D4—81no—
211C3×D4—81no—
212C3×D4—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—82no—
250C2×Dic3—82no—
251C3×C8—81no—
252C3⋊C8—81no—
253C24—21no—
254C24D3×D411yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—122no—
284C22×C4—122no—
285C22×C4—122no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—122no—
301C4⋊C4—124no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—124no—
305C22⋊C4—124no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—62no—
309C42—62no—
310C42—62no—
311C42—62no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3#38611yescharacteristic
326C22×C3#41012yes—
327C22×C3#41611yescharacteristic
328C22×C3#41811yescharacteristic
329C22×C3#42112yes—
330C22×C3#44911yescharacteristic
331C22×C3#44911yescharacteristic
332C22×C3#48211yescharacteristic
333C22×C3#48211yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—82no—
339C3×C4—81no—
340Dic3—42no—
341Dic3—42no—
342Dic3—82no—
343Dic3—82no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23D3×C22⋊C411yescharacteristic
352C23#88311yescharacteristic
353C23#91111yescharacteristic
354C23#91211yescharacteristic
355C23#91411yescharacteristic
356C23#91611yescharacteristic
357C23#93711yescharacteristic
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—242no—
397C2×C4—242no—
398C2×C4#67⋊C411yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4C4×#6711yescharacteristic
401C2×C4#93811yescharacteristic
402C8—81no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—161no—
409C2×C3#151412yes—
410C2×C3#152112yes—
411C2×C3#187511yescharacteristic
412C2×C3#189711yescharacteristic
413C2×C3#189911yescharacteristic
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22#474012yes—
428C22#477011yescharacteristic
429C22#477211yescharacteristic
430C22#477911yescharacteristic
431C22#478411yescharacteristic
432C22#482112yes—
433C22#495611yescharacteristic
434C22#515811yescharacteristic
435C22#517211yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—81no—
442C4—122no—
443C4—122no—
444C4—242no—
445C4—242no—
446C3#915711yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—161no—
452C2#6828112yes—
453C2#6828312yes—
454C2#7002911yescharacteristic
455C2#7020111yescharacteristic
456C2#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#9157
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×#4651C2semidirect
5. #69910C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
14. #5221C4semidirect
21. #4972C4semidirect
446. C3#9157semidirect

Maximal subgroups

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

subgroupindex
#826992
C2×#46512
C2×C3×#3272
C2×#45372
#699102
#676752
#676122
#91573