Finite Group Databasebeta

← #10690077

Subgroups — #10690077

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83296C211yesmaximal, characteristic
2#79012C211yesmaximal, characteristic, complemented
3C3×#1480C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#70754C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8995—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#950—21no—
11C2×#901—21no—
12C4×#235C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5233—21no—
16#5102—21no—
17#5090C411yescharacteristic
18#5083C411yescharacteristic
19#5075C411yescharacteristic
20#5072C411yescharacteristic
21#5057—21no—
22#5054—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4839—22no—
27#4837—21no—
28D3×C4⋊C8—21no—
29C3×D4⋊C8—21no—
30C3×D4⋊C8—21no—
31D4⋊(C3⋊C8)—22no—
32#4484—21no—
33#4483—21no—
34#3035—31no—
35#2906—31no—
36#1492—31no—
37#1480D311yescharacteristic, complemented
38#1476—31no—
39#1301—31no—
40#1277—31no—
41C2×C3×#41—21no—
42C2×#235D411yescharacteristic
43C22×C3×C8—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C42C2311yescharacteristic
52#950—42no—
53#950—42no—
54C2×#216—21no—
55C2×#215—21no—
56C2×#214—21no—
57C2×#213—21no—
58C4×#67—21no—
59C4×#67—21no—
60#926D411yescharacteristic
61C2×#208—21no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64#901—42no—
65#901—42no—
66C2×ΓL(1,𝔽25)—22no—
67C2×D3×C8—22no—
68C4×D12C2×C411yescharacteristic
69C4×D12D411yescharacteristic
70#872—21no—
71C42×D3—21no—
72C4×Dic6C2×C411yescharacteristic
73C4×Dic6D411yescharacteristic
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D411yescharacteristic
77C3×C4⋊C8D411yescharacteristic
78C3×C22⋊C8—21no—
79C3×C4×C8—21no—
80#835—21no—
81#835—21no—
82#832—21no—
83#832—21no—
84Dic3⋊C8—22no—
85Dic3⋊C8—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4⋊(C3⋊C8)C2×C411yescharacteristic
90C4⋊(C3⋊C8)D412yes—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×#135—61no—
94C2×#131—61no—
95#532C2×D311yescharacteristic
96C4×#41—31no—
97#464—61no—
98#463—61no—
99#462—31no—
100#461—31no—
101#460—31no—
102#459—31no—
103#431C2×D311yescharacteristic
104#424—62no—
105#424—61no—
106#423—31no—
107#423—61no—
108C2×C4⋊C8—31no—
109C2×C4⋊C8—61no—
110C2×C4⋊C8C2×D311yescharacteristic
111#416—61no—
112C2×C4×C8—31no—
113Q8⋊C8—61no—
114D4⋊C8—21no—
115D4⋊C8—21no—
116D4⋊C8—61no—
117D4⋊C8—62no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C2×C3×Q8—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132#235—21no—
133#235—21no—
134#235D812yes—
135#235SD1612yes—
136C2×D12C22⋊C411yescharacteristic
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×Dic6C22⋊C411yescharacteristic
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C22×C411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154#217—21no—
155#216—42no—
156#215—42no—
157#214—42no—
158#213—42no—
159(C2×D3)⋊C4—21no—
160(C2×D3)⋊C4—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178ΓL(1,𝔽25)—44no—
179D3×C8—44no—
180C2×#41—21no—
181C2×#41—31no—
182#135—122no—
183#135—122no—
184C2×Q16—61no—
185C2×SD16—61no—
186C2×SD16—61no—
187C2×D8—61no—
188#131—122no—
189#131—122no—
190C2×M16—62no—
191C2×M16—61no—
192C22×C8—21no—
193C22×C8—62no—
194C22×C8—61no—
195C22×C8—61no—
196C4⋊Q8C4×D311yescharacteristic
197#127C4×D311yescharacteristic
198#124—21no—
199#121—21no—
200#121—21no—
201C4×Q8—31no—
202C4×Q8—31no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—31no—
206C4×D4—31no—
207C4×D4—61no—
208C4×D4—61no—
209#117—31no—
210#117—61no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—31no—
216C4⋊C8—31no—
217C4⋊C8—32no—
218C4⋊C8—62no—
219C4⋊C8—62no—
220C4⋊C8—61no—
221C4⋊C8—61no—
222C4⋊C8#6711yescharacteristic
223C4⋊C8#6711yescharacteristic
224#103—61no—
225#103—61no—
226#102—61no—
227#102—61no—
228C22⋊C8—21no—
229C4×C8—21no—
230C4×C8—61no—
231C4×C8—61no—
232C23×C3—41no—
233C23×C3#12311yescharacteristic
234C22×D3—21no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×D4—42no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—81no—
243C3×D4—81no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4C2×D811yescharacteristic
257C2×C3×C4C2×SD1611yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4#13711yescharacteristic
260#67—22no—
261#67—22no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265D12—21no—
266D12#10212yes—
267C4×D3—22no—
268C4×D3—22no—
269C4×D3—41no—
270C4×D3—41no—
271C4×D3—81no—
272Dic6—21no—
273Dic6#10212yes—
274C3×C8—41no—
275C3×C8—41no—
276C3×C8—81no—
277C3⋊C8—42no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—42no—
281#41—32no—
282#41—32no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—61no—
288#41—61no—
289C2×Q8—21no—
290C2×Q8—31no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—31no—
295C2×D4—41no—
296C2×D4—61no—
297C22×C4—41no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4#23511yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4#23711yescharacteristic
304Q16—122no—
305SD16—122no—
306SD16—122no—
307D8—122no—
308M16—124no—
309M16—121no—
310M16—121no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—62no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—62no—
320C2×C8—124no—
321C2×C8—121no—
322C2×C8—121no—
323C2×C8—121no—
324C2×C8—121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C42—61no—
337C42—61no—
338C42—61no—
339C42C2×C4×D311yescharacteristic
340C42D3×D411yescharacteristic
341C42D3×D411yescharacteristic
342C42C2×#6711yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3#38611yescharacteristic
349C22×C3#48011yescharacteristic
350C22×C3#48111yescharacteristic
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4C2×#10211yescharacteristic
359C3×C4#41711yescharacteristic
360C3×C4#44311yescharacteristic
361C3×C4#44411yescharacteristic
362C3×C4#44711yescharacteristic
363C3×C4#44811yescharacteristic
364Dic3—22no—
365Dic3—41no—
366Dic3—41no—
367C23—41no—
368C23—61no—
369C23#88311yescharacteristic
370Q8—32no—
371Q8—41no—
372Q8—41no—
373Q8—61no—
374D4—32no—
375D4—42no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—62no—
381D4—62no—
382D4—61no—
383D4—81no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—61no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—241no—
399C2×C4D3×C22⋊C411yescharacteristic
400C2×C4#67⋊C411yescharacteristic
401C2×C4#88411yescharacteristic
402C2×C4D3×D811yescharacteristic
403C2×C4#91111yescharacteristic
404C2×C4D3×SD1611yescharacteristic
405C2×C4#91511yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93711yescharacteristic
408C2×C4#93811yescharacteristic
409C8—41no—
410C8—41no—
411C8—81no—
412C8—122no—
413C8—122no—
414C8—121no—
415C8—121no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—161no—
419C2×C3#187411yescharacteristic
420C2×C3#187611yescharacteristic
421C2×C3#188111yescharacteristic
422D3—42no—
423C22—21no—
424C22—21no—
425C22—62no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—121no—
430C22—121no—
431C22#477411yescharacteristic
432C22#477711yescharacteristic
433C22#495611yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—62no—
438C4—81no—
439C4—121no—
440C4—121no—
441C4D3×#10211yescharacteristic
442C4#477111yescharacteristic
443C4#514411yescharacteristic
444C4#514511yescharacteristic
445C4#515611yescharacteristic
446C4#517011yescharacteristic
447C3#899511yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—122no—
451C2—161no—
452C2#7001111yescharacteristic
453C2#7020011yescharacteristic
454C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1480C2
socleC22×C3#386
prime radical#83296C2
cosocleC2×C3×C42C23
hypercenter#1480D3
hypocenterC3#8995
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #79012C2semidirect
3. C3×#1480C2semidirect
4. #71802C2semidirect
5. #70754C2semidirect
6. #67745C2semidirect
7. #67703C2semidirect
37. #1480D3semidirect
447. C3#8995semidirect

Maximal subgroups

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

subgroupindex
#832962
#790122
C3×#14802
#718022
#707542
#677452
#677032
#89953