Finite Group Databasebeta

← #10778366

Subgroups — #10778366

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic, complemented
2#83334C211yesmaximal, characteristic, complemented
3#83310C211yesmaximal, characteristic
4#83275C211yesmaximal, characteristic, complemented
5#71830C211yesmaximal, characteristic, complemented
6#71139C212yesmaximal, complemented
7#71134C212yesmaximal, complemented
8#71043C212yesmaximal, complemented
9#71038C212yesmaximal, complemented
10#70432C211yesmaximal, characteristic, complemented
11C2×#4480C211yesmaximal, characteristic, complemented
12#19200—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C3×#532C2211yescharacteristic
15C2×#948—21no—
16#5477C2211yescharacteristic
17C3×#476C2212yes—
18C3×#475—21no—
19C3×#465—22no—
20C3×#461C2212yes—
21C3×#460—21no—
22C2×C3×C4⋊C8C2211yescharacteristic
23#5234—21no—
24#5093C2211yescharacteristic
25#5093C2211yescharacteristic
26#5088C2211yescharacteristic
27#5088C2211yescharacteristic
28#5087—21no—
29#5072—21no—
30#5068—21no—
31#5068—21no—
32#5038—22no—
33#5015C2212yes—
34#5012—21no—
35#5012C2212yes—
36D4×C3⋊C8—21no—
37D4×C3⋊C8C2212yes—
38#5007C2212yes—
39#5002—21no—
40#5002—21no—
41#4924C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4547C2212yes—
45#4546C2212yes—
46#4486C2212yes—
47#4485C2212yes—
48#4480C2212yes—
49#4480C2212yes—
50#3246D311yescharacteristic, complemented
51#3245—31no—
52#3131—31no—
53#2931—31no—
54#1703—32no—
55#1701—32no—
56#1679—32no—
57#1677—32no—
58C2×#334—31no—
59#1550—31no—
60#1547—31no—
61C2×C3×#41—21no—
62C2×#235—21no—
63C2×C3×SD16—22no—
64C22×C3×C8—21no—
65C3×C4⋊Q8D411yescharacteristic
66C3×C4⋊Q8C2311yescharacteristic
67C3×C4⋊Q8C2311yescharacteristic
68C3×#127D411yescharacteristic
69C3×#125—21no—
70C3×#124—21no—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×#121—21no—
74C3×#121D412yes—
75C3×C4×Q8—21no—
76C3×C4×Q8—21no—
77C3×C4×D4—21no—
78C3×C4×D4C2312yes—
79C3×#117—21no—
80C2×C3×C42C2311yescharacteristic
81#948—44no—
82C2×#213—22no—
83#930—21no—
84#930—21no—
85C2×#208—21no—
86C2×#208—22no—
87C22×C3⋊C8—21no—
88C22×C3⋊C8—22no—
89C22×C3⋊C8D411yescharacteristic
90C22×C3⋊C8D411yescharacteristic
91#875—21no—
92#874C2311yescharacteristic
93C4×D12—21no—
94C4×D12—21no—
95#869C2311yescharacteristic
96C3×#106—22no—
97C3×C4⋊C8—21no—
98C3×C4⋊C8C2312yes—
99C3×#103—22no—
100C3×#102—22no—
101#832—22no—
102#830—22no—
103#830—22no—
104#810—21no—
105#810—21no—
106#810—21no—
107#810—21no—
108#809—22no—
109#809—21no—
110#809—21no—
111#809—21no—
112#809—21no—
113#807—22no—
114C4⋊(C3⋊C8)—21no—
115C4⋊(C3⋊C8)—21no—
116C4⋊(C3⋊C8)C2312yes—
117C4⋊(C3⋊C8)C2312yes—
118C4×C3⋊C8D411yescharacteristic
119C4×C3⋊C8D411yescharacteristic
120C4×C3⋊C8C2311yescharacteristic
121C4×C3⋊C8C2311yescharacteristic
122C2×#131—61no—
123#533C2×D311yescharacteristic
124#532—31no—
125#532C2×D311yescharacteristic
126#488—61no—
127#488—61no—
128#487—31no—
129#487—31no—
130#486—31no—
131#486—31no—
132#476C2×D312yes—
133#475—21no—
134#474—32no—
135#474—61no—
136#466—62no—
137#465—22no—
138#461C2×D312yes—
139#460—21no—
140#459—32no—
141#459—61no—
142#436—32no—
143#436—61no—
144C8×D4—32no—
145C8×D4—61no—
146#423—31no—
147C2×C4⋊C8—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149#416—61no—
150#416—61no—
151#408—61no—
152C2×C4×C8—31no—
153#334—32no—
154#334—32no—
155#333—32no—
156#332—32no—
157#332—32no—
158#331—32no—
159C3×#41—42no—
160C3×#41—42no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4#4111yescharacteristic
172C2×#67—41no—
173#235—42no—
174#235—42no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×C4×D3—41no—
179C2×Dic6—21no—
180C3×SD16—82no—
181C2×C3×C8—22no—
182C2×C3×C8—81no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—41no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—41no—
194C3×C42—41no—
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C3×C42#4111yescharacteristic
198C3×C42C2411yescharacteristic
199#213—82no—
200(C2×D3)⋊C4—41no—
201(C2×D3)⋊C4—41no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204#208—44no—
205#208—42no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—44no—
211C2×C3⋊C8—81no—
212C2×C3⋊C8C2×D411yescharacteristic
213C2×C3⋊C8C2×D411yescharacteristic
214C2×C3⋊C8C2×D411yescharacteristic
215C2×C3⋊C8C2×D411yescharacteristic
216C2×#41—21no—
217C2×#41—61no—
218C2×SD16—22no—
219C2×D8—62no—
220#131—124no—
221C2×M16—62no—
222C2×M16—61no—
223C22×C8—21no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—62no—
227C22×C8—61no—
228C4⋊Q8—31no—
229C4⋊Q8#6711yescharacteristic
230C4⋊Q8C22×D311yescharacteristic
231C4⋊Q8C22×D311yescharacteristic
232#127—31no—
233#127#6711yescharacteristic
234#125—21no—
235#124—21no—
236#124—61no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—21no—
240#121—61no—
241#121—61no—
242#121#6712yes—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×D4—21no—
246C4×D4—61no—
247C4×D4—61no—
248C4×D4C22×D312yes—
249#117—21no—
250C2×C42C22×D311yescharacteristic
251#107—62no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—32no—
255C4⋊C8—32no—
256C4⋊C8—61no—
257C4⋊C8—61no—
258C4⋊C8C22×D312yes—
259#103—22no—
260#103—61no—
261#103—61no—
262#103—61no—
263#103—61no—
264#102—22no—
265#102—61no—
266#102—61no—
267#102—61no—
268#102—61no—
269#102—62no—
270#102—62no—
271C22⋊C8—62no—
272C22⋊C8—62no—
273C4×C8—31no—
274C4×C8—31no—
275C4×C8—31no—
276C4×C8—31no—
277C23×C3—22no—
278C23×C3#12111yescharacteristic
279C22×D3—41no—
280C3×Q8—42no—
281C3×Q8—41no—
282C3×Q8—41no—
283C3×Q8—81no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—42no—
287C3×D4—44no—
288C3×D4—42no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4#12111yescharacteristic
298C2×C3×C4#12111yescharacteristic
299C2×C3×C4#12111yescharacteristic
300C2×C3×C4#13512yes—
301C2×C3×C4AGL(1,ℤ/8)12yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305#67—82no—
306C2×Dic3—41no—
307D12—41no—
308D12—41no—
309D12—41no—
310D12—41no—
311D12—81no—
312C4×D3—82no—
313Dic6—41no—
314Dic6—41no—
315C3×C8—42no—
316C3⋊C8—21no—
317C3⋊C8—21no—
318C3⋊C8—21no—
319C3⋊C8—21no—
320C3⋊C8—42no—
321C3⋊C8—42no—
322#41—42no—
323#41—42no—
324#41—122no—
325#41—122no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—61no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—121no—
337C2×D4C2×#6712yes—
338C22×C4—22no—
339C22×C4—121no—
340C22×C4D3×D411yescharacteristic
341C22×C4#23711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343SD16—82no—
344D8—242no—
345M16—124no—
346M16—122no—
347C2×C8—22no—
348C2×C8—31no—
349C2×C8—31no—
350C2×C8—31no—
351C2×C8—31no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—62no—
356C2×C8—81no—
357C2×C8—124no—
358C2×C8—241no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4C2×#6712yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—41no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C42—41no—
375C42D3×D411yescharacteristic
376C42#23711yescharacteristic
377C42C2×#6711yescharacteristic
378C42C23×D311yescharacteristic
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3#46312yes—
385C22×C3C2×#12111yescharacteristic
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—42no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4#46312yes—
396C3×C4C2×#13511yescharacteristic
397C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C3×C4#57711yescharacteristic
399C3×C4#57711yescharacteristic
400Dic3—81no—
401C23—22no—
402C23—121no—
403C23#93911yescharacteristic
404Q8—42no—
405Q8—41no—
406Q8—41no—
407Q8—81no—
408Q8—121no—
409Q8—121no—
410D4—22no—
411D4—22no—
412D4—42no—
413D4—44no—
414D4—42no—
415D4—121no—
416D4—121no—
417D4—121no—
418D4—121no—
419D4—241no—
420D4—242no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—121no—
430C2×C4—242no—
431C2×C4#91412yes—
432C2×C4#91612yes—
433C2×C4#93911yescharacteristic
434C2×C4#93911yescharacteristic
435C2×C4#93911yescharacteristic
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×#23711yescharacteristic
438C2×C4C22×#6711yescharacteristic
439C8—42no—
440C8—61no—
441C8—61no—
442C8—61no—
443C8—61no—
444C8—122no—
445C8—122no—
446C2×C3—22no—
447C2×C3—82no—
448C2×C3C2×#46311yescharacteristic
449C2×C3#306011yescharacteristic
450C2×C3#306711yescharacteristic
451D3—161no—
452C22—21no—
453C22—21no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22#516512yes—
461C22C2×#93911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—241no—
469C4#516512yes—
470C4C2×#91411yescharacteristic
471C4C2×#91611yescharacteristic
472C4#583611yescharacteristic
473C4#583711yescharacteristic
474C3#1920011yescharacteristic, complemented, Sylow
475C2—22no—
476C2—82no—
477C2—481no—
478C2#8342011yescharacteristic
479C2C2×#516511yescharacteristic
480C2#8400111yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3246C2
socleC22×C3C2×#121
prime radical#83310C2
hypercenter#3246D3
hypocenterC3#19200
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
1. C3×#3246C2semidirect
2. #83334C2semidirect
4. #83275C2semidirect
5. #71830C2semidirect
6. #71139C2semidirect
7. #71134C2semidirect
8. #71043C2semidirect
9. #71038C2semidirect
10. #70432C2semidirect
11. C2×#4480C2semidirect
50. #3246D3semidirect
474. C3#19200semidirect

Maximal subgroups

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

subgroupindex
C3×#32462
#833342
#833102
#832752
#718302
#711392
#711342
#710432
#710382
#704322
C2×#44802
#192003