Finite Group Databasebeta

← #10691669

Subgroups — #10691669

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2977C211yesmaximal, characteristic, complemented
2#82929C211yesmaximal, characteristic
3#71708C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic, complemented
5#69921C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9264—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#435—21no—
12C3×#435—21no—
13C3×C8×D4—21no—
14C3×C8×D4—21no—
15C3×#433—21no—
16C3×#408—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18C2×C3×C22⋊C8C2211yescharacteristic
19#5238—21no—
20C2×C3⋊#95C2211yescharacteristic
21#5207C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5048—21no—
24#5046C411yescharacteristic, complemented
25#5044C411yescharacteristic, complemented
26#5038—21no—
27#5036C411yescharacteristic, complemented
28#5034C411yescharacteristic, complemented
29#5003—21no—
30#4538—21no—
31#4537—21no—
32#4470—21no—
33#4470—21no—
34#4468—21no—
35#4468—21no—
36#3188—31no—
37#2977D311yescharacteristic, complemented
38#1504—31no—
39C2×#327—31no—
40#1471—31no—
41#1298—31no—
42#1280—31no—
43C2×C3×#41D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C23×Dic3—21no—
47C22×D12C2×C411yescharacteristic
48C2×C3×M16—21no—
49C22×C3×C8—21no—
50C22×C3×C8—21no—
51C22×C3×C8—21no—
52C3×#123—21no—
53C3×C22⋊Q8C2×C411yescharacteristic
54C3×C22⋊Q8D411yescharacteristic
55C3×#121C2×C411yescharacteristic
56C3×#121D411yescharacteristic
57C3×#120—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×#117D411yescharacteristic
61C2×#215—21no—
62C2×#213—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3C2×C411yescharacteristic
69C2×#208—21no—
70C22×C3⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D411yescharacteristic
78C3×C22⋊C8D411yescharacteristic
79C3×C22⋊C8D411yescharacteristic
80C3×C22⋊C8D411yescharacteristic
81C3×#97—21no—
82C3×C4×C8—21no—
83#835—21no—
84#832—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830C2×C411yescharacteristic
91#830C2×C411yescharacteristic
92#809—21no—
93#809—21no—
94#808—21no—
95#807—21no—
96#525C2×D311yescharacteristic
97C2×#121—31no—
98#484—31no—
99#482—61no—
100#481—31no—
101#466—61no—
102#465—61no—
103#450—31no—
104#449—61no—
105#447—31no—
106#435—21no—
107#435—21no—
108C8×D4—21no—
109C8×D4—21no—
110#433—21no—
111#408—21no—
112#407—31no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8C2×D311yescharacteristic
115C2×C22⋊C8C2×D311yescharacteristic
116C2×#95—31no—
117#328—61no—
118#327—61no—
119#327—61no—
120#327—61no—
121#323—61no—
122#323—61no—
123C24×C3#4111yescharacteristic
124C23×D3—21no—
125C3×#41—81no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4C22⋊C411yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4—81no—
133C22×C3×C4C22×C411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—81no—
142C2×#67—81no—
143C22×Dic3—22no—
144C22×Dic3—21no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—81no—
148C22×Dic3—81no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—81no—
152C3×M16—81no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—81no—
158C2×C3×C8—81no—
159C2×C3×C8—81no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4C22⋊C411yescharacteristic
163C3×C4⋊C4C22⋊C411yescharacteristic
164C3×C22⋊C4—21no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—21no—
167C3×C22⋊C4—21no—
168C3×C22⋊C4—21no—
169C3×C42—21no—
170#215—81no—
171#213—81no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176#208—81no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—81no—
181C2×#41#6711yescharacteristic
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—61no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—61no—
187C2×D8—61no—
188C2×M16—21no—
189C2×M16—61no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—61no—
194#123—21no—
195C22⋊Q8C4×D311yescharacteristic
196C22⋊Q8D1211yescharacteristic
197#121—61no—
198#121—61no—
199#121—61no—
200#121—61no—
201#121C4×D311yescharacteristic
202#121D1211yescharacteristic
203#120—21no—
204C4×D4—21no—
205C4×D4—21no—
206#117#6711yescharacteristic
207C2×C4⋊C4—31no—
208C2×C22⋊C4—61no—
209#107—61no—
210#106—61no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213#103—61no—
214#102—61no—
215#102—61no—
216#102—61no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—31no—
222C22⋊C8—31no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8#6711yescharacteristic
227C22⋊C8#6711yescharacteristic
228C22⋊C8#6711yescharacteristic
229C22⋊C8#6711yescharacteristic
230#97—21no—
231C4×C8—21no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3C4×D411yescharacteristic
241C23×C3#12111yescharacteristic
242C23×C3#12311yescharacteristic
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—81no—
247C22×D3—81no—
248C3×Q8—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—81no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4C2×C22⋊C411yescharacteristic
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270#67—82no—
271#67—81no—
272#67—81no—
273C2×Dic3—42no—
274C2×Dic3—41no—
275C2×Dic3—81no—
276C2×Dic3—82no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—82no—
280D12—41no—
281D12—41no—
282D12—81no—
283C3×C8—41no—
284C3×C8—41no—
285C3×C8—41no—
286C3×C8—41no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C3⋊C8—81no—
290C24—61no—
291C24#23511yescharacteristic
292#41—81no—
293C2×Q8(C2×D3)⋊C411yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—241no—
302C2×D4—241no—
303C2×D4—241no—
304C2×D4(C2×D3)⋊C411yescharacteristic
305C22×C4—21no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—81no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—241no—
312C22×C4—241no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4C2×D1211yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319SD16—241no—
320D8—241no—
321M16—81no—
322M16—241no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—81no—
330C2×C8—81no—
331C2×C8—81no—
332C2×C8—121no—
333C2×C8—241no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4(C2×D3)⋊C411yescharacteristic
339C4⋊C4(C2×D3)⋊C411yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C42—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3#38611yescharacteristic
363C22×C3#44311yescharacteristic
364C22×C3#44411yescharacteristic
365C22×C3#47111yescharacteristic
366C22×C3#47211yescharacteristic
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—81no—
370C2×D3—161no—
371C2×D3—161no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379Dic3—82no—
380Dic3—81no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—41no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—241no—
391C23—241no—
392C23C4×#6711yescharacteristic
393C23#92911yescharacteristic
394C23#93011yescharacteristic
395Q8—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—81no—
402D4—121no—
403D4—121no—
404D4—241no—
405D4—242no—
406D4—241no—
407D4—241no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—122no—
420C2×C4—121no—
421C2×C4—241no—
422C2×C4—242no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4—242no—
426C2×C4C2×(C2×D3)⋊C411yescharacteristic
427C2×C4C4×#6711yescharacteristic
428C2×C4#93011yescharacteristic
429C2×C4#95311yescharacteristic
430C8—41no—
431C8—41no—
432C8—41no—
433C8—41no—
434C8—121no—
435C8—121no—
436C8—241no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—41no—
442C2×C3—81no—
443C2×C3#187611yescharacteristic
444C2×C3#190711yescharacteristic
445C2×C3#191311yescharacteristic
446D3—161no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—241no—
462C22—241no—
463C22—241no—
464C22—481no—
465C22—481no—
466C22#514111yescharacteristic
467C22#514211yescharacteristic
468C22#514411yescharacteristic
469C22#514511yescharacteristic
470C22#521311yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—242no—
479C4—241no—
480C3#926411yescharacteristic, complemented, Sylow
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—481no—
488C2#7165611yescharacteristic
489C2#7201011yescharacteristic
490C2#7201811yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2977C2
socleC22×C3#386
prime radical#82929C2
cosocleC23×C3×C4C23
hypercenter#2977D3
hypocenterC3#9264
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2977

Complements

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

NH ≅ G/Ntype
1. C3×#2977C2semidirect
3. #71708C2semidirect
4. C2×#4470C2semidirect
5. #69921C2semidirect
6. #67680C2semidirect
7. #67674C2semidirect
24. #5046C4semidirect
25. #5044C4semidirect
27. #5036C4semidirect
28. #5034C4semidirect
37. #2977D3semidirect
480. C3#9264semidirect

Maximal subgroups

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

subgroupindex
C3×#29772
#829292
#717082
C2×#44702
#699212
#676802
#676742
#92643