Finite Group Databasebeta

← #10691648

Subgroups — #10691648

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2345C211yesmaximal, characteristic, complemented
2C2×#4968C211yesmaximal, characteristic
3#71554C211yesmaximal, characteristic, complemented
4#69509C211yesmaximal, characteristic
5#69456C211yesmaximal, characteristic
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9261—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#389—21no—
12C3×#387—22no—
13C3×#380—21no—
14C3×#380C2211yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#4968—22no—
22#4968—21no—
23#4968—21no—
24C2×#809C412yescomplemented
25C2×#808C412yescomplemented
26C2×#806—21no—
27C2×#806—21no—
28#4529—22no—
29#4451—22no—
30C2×#481—31no—
31#2345D311yescharacteristic, complemented
32#1964—31no—
33#1824—31no—
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C22×D12C2×C411yescharacteristic
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4C2×C412yes—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42—22no—
55C2×#217—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C22×C3⋊C8—21no—
64C22×C3⋊C8C2×C412yes—
65C3×#95—22no—
66C3×#95—22no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#809—22no—
75#809—22no—
76#808—22no—
77#808—22no—
78#806—42no—
79#806—42no—
80C2×#121—31no—
81C22×C4⋊C4C2×D311yescharacteristic
82#481—62no—
83#481—61no—
84#481—61no—
85C2×#106—32no—
86C2×#102—32no—
87C2×#99—61no—
88C2×#99—61no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8—31no—
91#389—21no—
92#387—22no—
93#380—21no—
94#380—31no—
95#380C2×D311yescharacteristic
96C4×C4⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4C2×D311yescharacteristic
99#340—62no—
100#327—62no—
101C24×C3C2×D411yescharacteristic
102C23×D3—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4—82no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119C2×#67—41no—
120C2×#67—41no—
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—41no—
124C22×Dic3—21no—
125C22×Dic3—41no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—42no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4—82no—
135C3×C4⋊C4C22⋊C412yes—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—21no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4SD1612yes—
143C3×C42—42no—
144C3×C42—82no—
145#217—41no—
146#217—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—42no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C22×D4—31no—
157C22×D4—61no—
158C23×C4—21no—
159C23×C4C22×D311yescharacteristic
160C22×C8—32no—
161C22×C8—61no—
162#121—61no—
163#121—61no—
164#121—61no—
165#121—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4C4×D312yes—
174C2×C4⋊C4D1212yes—
175C2×C4⋊C4#6711yescharacteristic
176C2×C4⋊C4#6711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4#6711yescharacteristic
181C2×C22⋊C4#6711yescharacteristic
182C2×C42—21no—
183C2×C42—22no—
184#106—62no—
185#106—62no—
186#102—62no—
187#102—62no—
188#99—122no—
189#99—122no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194#95—22no—
195#95—22no—
196#95—21no—
197#95—61no—
198#95#6712yes—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3C4×D411yescharacteristic
202C23×C3#12011yescharacteristic
203C23×C3#12111yescharacteristic
204C23×C3C2×SD1611yescharacteristic
205C23×C3#13511yescharacteristic
206C23×C3AGL(1,ℤ/8)11yescharacteristic
207C23×C3#13711yescharacteristic
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—81no—
212C22×D3—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
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—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×C3×C4C2×C22⋊C411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238#67—81no—
239#67—81no—
240#67—81no—
241#67—81no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248D12—42no—
249D12—81no—
250C3⋊C8—42no—
251C3⋊C8—81no—
252C24—61no—
253C24C2×#6711yescharacteristic
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—122no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—61no—
272C22×C4—82no—
273C22×C4—121no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4C2×D1211yescharacteristic
276C22×C4#23511yescharacteristic
277C22×C4#23511yescharacteristic
278C22×C4C2×#6711yescharacteristic
279C22×C4C2×#6711yescharacteristic
280C2×C8—62no—
281C2×C8—122no—
282C2×C8—122no—
283C2×C8—121no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—82no—
292C4⋊C4—122no—
293C4⋊C4(C2×D3)⋊C412yes—
294C4⋊C4(C2×D3)⋊C412yes—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4#21512yes—
305C42—42no—
306C42—82no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3C4×SD1611yescharacteristic
319C22×C3#44011yescharacteristic
320C22×C3#44811yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#46011yescharacteristic
323C22×C3#46411yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—161no—
328C2×D3—161no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—81no—
336C3×C4—82no—
337C3×C4#41012yes—
338Dic3—81no—
339Dic3—161no—
340C23—41no—
341C23—41no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—241no—
346C23—241no—
347C23C4×#6711yescharacteristic
348C23#93011yescharacteristic
349C23C2×#21511yescharacteristic
350C23#94211yescharacteristic
351C23#94911yescharacteristic
352C23#95011yescharacteristic
353C23#95311yescharacteristic
354D4—122no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4C2×(C2×D3)⋊C411yescharacteristic
388C2×C4C4×#6711yescharacteristic
389C2×C4#92911yescharacteristic
390C2×C4#93011yescharacteristic
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#188011yescharacteristic
400C2×C3#189911yescharacteristic
401C2×C3#190411yescharacteristic
402D3—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—481no—
417C22—481no—
418C22C4×#21511yescharacteristic
419C22#502611yescharacteristic
420C22#502711yescharacteristic
421C22#502811yescharacteristic
422C22#521311yescharacteristic
423C22#522711yescharacteristic
424C22#523811yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—82no—
433C4—241no—
434C4—481no—
435C4#500612yes—
436C3#926111yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—481no—
444C2#7159511yescharacteristic
445C2#7163311yescharacteristic
446C2#7165111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2345C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2345D3
hypocenterC3#9261
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
1. C3×#2345C2semidirect
3. #71554C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
24. C2×#809C4semidirect
25. C2×#808C4semidirect
31. #2345D3semidirect
436. C3#9261semidirect

Maximal subgroups

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

subgroupindex
C3×#23452
C2×#49682
#715542
#695092
#694562
#676252
#676172
#92613