Finite Group Databasebeta

← #10950833

Subgroups — #10950833

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78569C211yesmaximal, characteristic, complemented
2#78568C211yesmaximal, characteristic, complemented
3C3×#2108C211yesmaximal, characteristic, complemented
4#71444C211yesmaximal, characteristic, complemented
5#71443C211yesmaximal, characteristic, complemented
6#71414C211yesmaximal, characteristic, complemented
7#71413C211yesmaximal, characteristic, complemented
8#69081C211yesmaximal, characteristic, complemented
9#69078C211yesmaximal, characteristic, complemented
10#69077C211yesmaximal, characteristic, complemented
11#69069C211yesmaximal, characteristic, complemented
12C4×#800C211yesmaximal, characteristic, complemented
13#69059C211yesmaximal, characteristic, complemented
14#69058C211yesmaximal, characteristic, complemented
15C4×D48C211yesmaximal, characteristic, complemented
16#33744—31nomaximal, Sylow
17#5535—21no—
18#5534—21no—
19C2×C3×M32C2211yescharacteristic
20C22×C3×C16C2211yescharacteristic
21C3×#405C2211yescharacteristic
22#5135—21no—
23#5134—21no—
24#5116C2211yescharacteristic
25#5115C2211yescharacteristic
26#5115C2211yescharacteristic
27#5115C2211yescharacteristic
28C2×#800C2211yescharacteristic
29C2×#800C2211yescharacteristic
30C2×D48C2211yescharacteristic
31C2×D48C2211yescharacteristic
32#4714—21no—
33#4713—21no—
34#4711—21no—
35#4710—21no—
36C4×Dic12C2211yescharacteristic
37#4696C2211yescharacteristic
38#4695C2211yescharacteristic
39C4×D24C2211yescharacteristic
40C4×D24C2211yescharacteristic
41C4×D24C2211yescharacteristic
42#4683C2211yescharacteristic
43#4682C2211yescharacteristic
44C3×#346C2211yescharacteristic
45C3×#346C2211yescharacteristic
46C3×C4×C16C2211yescharacteristic
47C3×C4×C16C2211yescharacteristic
48#4509C2211yescharacteristic
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4509C2211yescharacteristic
52#4509C2211yescharacteristic
53#4509C2211yescharacteristic
54#4505C2211yescharacteristic
55#4505C2211yescharacteristic
56#4504C2211yescharacteristic
57#4504C2211yescharacteristic
58#4503C2211yescharacteristic
59#4503C2211yescharacteristic
60#3176—31no—
61#3175—31no—
62#2255—31no—
63#2250—31no—
64#2242—31no—
65#2240—31no—
66#2220—31no—
67#2218—31no—
68#2217—31no—
69#2215—31no—
70#2177—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×D16—31no—
74#2108D311yescharacteristic, complemented
75C2×C3×M16D411yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×#117D411yescharacteristic
78#930—21no—
79#930—21no—
80#930—21no—
81#924—21no—
82C2×Dic12C2311yescharacteristic
83C2×D24—21no—
84C2×D24C2311yescharacteristic
85C2×D24C2311yescharacteristic
86C2×D24C2311yescharacteristic
87C2×#204—21no—
88#898—41no—
89#896—41no—
90#894—41no—
91#894—41no—
92#885—41no—
93#884—41no—
94#884—41no—
95#883—41no—
96#875—21no—
97#874—21no—
98C4×D12—21no—
99C4×D12—21no—
100C4×D12—21no—
101#870—21no—
102#869—21no—
103C4×Dic6—21no—
104C3×M32—22no—
105C2×C3×C16—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C2×C3×C16C2311yescharacteristic
109C2×C3×C16C2311yescharacteristic
110C3×#97D411yescharacteristic
111C3×#97D411yescharacteristic
112C3×C4×C8C2311yescharacteristic
113C3×C4×C8C2311yescharacteristic
114#821—21no—
115#821—21no—
116#821—21no—
117#821—21no—
118#821—21no—
119#821—21no—
120#818—21no—
121#818C2311yescharacteristic
122#818C2311yescharacteristic
123#818C2311yescharacteristic
124#818C2311yescharacteristic
125#817—21no—
126#816—21no—
127#816—21no—
128#800—22no—
129#800—22no—
130D48—22no—
131D48—22no—
132#542—61no—
133#540—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×D16—31no—
137C2×D16—31no—
138C2×M32C2×D311yescharacteristic
139C22×C16C2×D311yescharacteristic
140#500—31no—
141#499—31no—
142#495—31no—
143#493—31no—
144#490—61no—
145#489—61no—
146#469—61no—
147#468—61no—
148#467—31no—
149#466—31no—
150#466—31no—
151#466—31no—
152#442—61no—
153#440—61no—
154C4×Q16—31no—
155C4×D8—31no—
156C4×D8—31no—
157C4×D8—31no—
158#405C2×D311yescharacteristic
159#367—31no—
160#367—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#346C2×D311yescharacteristic
172#346C2×D311yescharacteristic
173C4×C16C2×D311yescharacteristic
174C4×C16C2×D311yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C2×#67—41no—
177C2×#67—41no—
178C2×#67—41no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—41no—
183C2×D12—41no—
184C2×C4×D3—41no—
185C2×C4×D3—41no—
186C2×C4×D3—41no—
187C2×Dic6—21no—
188C2×Dic6—41no—
189C3×M16—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2411yescharacteristic
196C3×C4⋊C4D811yescharacteristic
197C3×C4⋊C4D811yescharacteristic
198C3×C22⋊C4D811yescharacteristic
199C3×C22⋊C4D811yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202#217—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219Dic3⋊C4—41no—
220C4×Dic3—41no—
221Dic12—21no—
222Dic12—21no—
223D24—21no—
224D24—21no—
225D24—21no—
226D24—21no—
227D24—21no—
228D24—21no—
229D24—42no—
230#204—42no—
231C3×C16—21no—
232C3×C16—21no—
233C3×C16#4112yes—
234C3×C16#4112yes—
235C2×Q16—31no—
236C2×SD16—61no—
237C2×D8—31no—
238C2×D8—31no—
239C2×D8—31no—
240C2×D8—61no—
241C2×M16D1211yescharacteristic
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—61no—
244#127—61no—
245#126—121no—
246#125—61no—
247#124—61no—
248#124—121no—
249#123—121no—
250#123—121no—
251#123—121no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—61no—
255#121—61no—
256#121—61no—
257#121—121no—
258#121—121no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117D1211yescharacteristic
264SD32—62no—
265SD32—62no—
266D16—62no—
267D16—62no—
268M32—22no—
269C2×C16—22no—
270C2×C16C22×D311yescharacteristic
271C2×C16C22×D311yescharacteristic
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107—31no—
277#107—31no—
278#107—61no—
279#106—61no—
280#103—61no—
281#103—61no—
282#102—61no—
283#102—61no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97D1211yescharacteristic
289#97D1211yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3C2×D811yescharacteristic
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4C2×D811yescharacteristic
299C2×C3×C4C2×D811yescharacteristic
300C2×C3×C4C2×D811yescharacteristic
301C2×C3×C4C2×D811yescharacteristic
302C2×C3×C4C2×D811yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304#67—161no—
305#67—161no—
306#67—161no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312D12—41no—
313D12—41no—
314D12—41no—
315D12—41no—
316D12—41no—
317D12—41no—
318D12—161no—
319D12—161no—
320C4×D3—161no—
321C4×D3—161no—
322C4×D3—161no—
323Dic6—41no—
324Dic6—41no—
325Dic6—161no—
326C3×C8—21no—
327C3×C8—21no—
328C3×C8#13512yes—
329C3×C8C2×#4111yescharacteristic
330C3×C8C2×#4111yescharacteristic
331C2×Q8—61no—
332C2×Q8—121no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—121no—
337C2×D4—121no—
338C2×D4—121no—
339C2×D4—121no—
340C2×D4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4C2×D1211yescharacteristic
345Q16—61no—
346Q16—61no—
347SD16—122no—
348D8—61no—
349D8—61no—
350D8—61no—
351D8—61no—
352D8—61no—
353D8—61no—
354D8—122no—
355M16—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×D1211yescharacteristic
359C2×C8C2×D1211yescharacteristic
360C2×C8C2×D1211yescharacteristic
361C2×C8C23×D311yescharacteristic
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4D2411yescharacteristic
372C4⋊C4D2411yescharacteristic
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4D2411yescharacteristic
383C22⋊C4D2411yescharacteristic
384C42—121no—
385C42C2×D1211yescharacteristic
386C42C2×D1211yescharacteristic
387C16—21no—
388C16—21no—
389C16#23512yes—
390C16#23512yes—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3C22×D811yescharacteristic
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4#53211yescharacteristic
410C3×C4C2×#13511yescharacteristic
411Dic3—81no—
412Dic3—81no—
413Dic3—81no—
414Dic3—81no—
415Dic3—81no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23C2×D2411yescharacteristic
420Q8—121no—
421Q8—121no—
422Q8—481no—
423D4—121no—
424D4—121no—
425D4—121no—
426D4—121no—
427D4—121no—
428D4—121no—
429D4—481no—
430D4—481no—
431D4—481no—
432D4—481no—
433D4—481no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—481no—
442C2×C4—481no—
443C2×C4—481no—
444C2×C4C2×D2411yescharacteristic
445C2×C4C2×D2411yescharacteristic
446C2×C4C2×D2411yescharacteristic
447C2×C4C2×D2411yescharacteristic
448C2×C4C2×D2411yescharacteristic
449C2×C4C22×D1211yescharacteristic
450C8—21no—
451C8—21no—
452C8#90412yes—
453C8C2×#23511yescharacteristic
454C8C2×#23511yescharacteristic
455C2×C3—41no—
456C2×C3#316911yescharacteristic
457C2×C3#341511yescharacteristic
458C2×C3#341611yescharacteristic
459D3—161no—
460D3—161no—
461D3—161no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—241no—
466C22—241no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22—241no—
474C22C22×D2411yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—241no—
481C4—241no—
482C4—241no—
483C4—241no—
484C4—241no—
485C4#547711yescharacteristic
486C4C2×#90411yescharacteristic
487C3#3374411yescharacteristic, complemented, Sylow
488C2—41no—
489C2—481no—
490C2—481no—
491C2—481no—
492C2#7847011yescharacteristic
493C2#8201311yescharacteristic
494C2#8201411yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2108C2
socleC22×C3C22×D8
hypercenter#2108D3
hypocenterC3#33744
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
1. #78569C2semidirect
2. #78568C2semidirect
3. C3×#2108C2semidirect
4. #71444C2semidirect
5. #71443C2semidirect
6. #71414C2semidirect
7. #71413C2semidirect
8. #69081C2semidirect
9. #69078C2semidirect
10. #69077C2semidirect
11. #69069C2semidirect
12. C4×#800C2semidirect
13. #69059C2semidirect
14. #69058C2semidirect
15. C4×D48C2semidirect
74. #2108D3semidirect
487. C3#33744semidirect

Maximal subgroups

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

subgroupindex
#785692
#785682
C3×#21082
#714442
#714432
#714142
#714132
#690812
#690782
#690772
#690692
C4×#8002
#690592
#690582
C4×D482
#337443