Finite Group Databasebeta

← #10690671

Subgroups — #10690671

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#932C211yesmaximal, characteristic
2#78447C211yesmaximal, characteristic, complemented
3C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
4#71601C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic, complemented
6#67706C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9097—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932C411yescharacteristic
11C4×#235C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5215C411yescharacteristic
15#5111—22no—
16#5018—21no—
17C4×#214—21no—
18#5014—21no—
19C4×#213—21no—
20#4966C411yescharacteristic
21#4964C411yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4688—21no—
26C4×ΓL(1,𝔽25)—21no—
27#4685—21no—
28#4679—21no—
29C3×D4⋊C8—21no—
30C3×D4⋊C8—21no—
31#4525—22no—
32#4484—21no—
33#4483—21no—
34#2965—31no—
35C4×AGL(1,ℤ/8)—31no—
36#1477—31no—
37C2×D4⋊C8D311yescharacteristic, complemented
38#1460—31no—
39#1283—31no—
40#1278—31no—
41C22×C3×D4C2×C411yescharacteristic
42C23×C3×C4—21no—
43C2×#235C2×C411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—21no—
54#932—22no—
55#932—22no—
56C2×#213—21no—
57C4×#67—21no—
58C4×#67—21no—
59#926C2×C411yescharacteristic
60C2×#208C2×C411yescharacteristic
61C2×#208C2×C411yescharacteristic
62C2×ΓL(1,𝔽25)—22no—
63C4×D12D411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67C4×Dic6D411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8D411yescharacteristic
73C3×C4×C8D411yescharacteristic
74#832—21no—
75#832—21no—
76#830—21no—
77#830D412yes—
78(C2×D3)⋊C8—22no—
79#815—22no—
80Dic3⋊C8—22no—
81#810—21no—
82#809—21no—
83#808—21no—
84#807—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×AGL(1,ℤ/8)—31no—
92C4×#41—31no—
93C2×C4×D4C2×D311yescharacteristic
94#446—61no—
95#442—61no—
96#440—61no—
97C4×SD16—61no—
98C4×D8—61no—
99#436—61no—
100#435—62no—
101#431—31no—
102#431—31no—
103#431—61no—
104#428—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106#418—31no—
107#417—31no—
108C4×M16—31no—
109C4×M16—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—61no—
112D4⋊C8—21no—
113D4⋊C8—21no—
114D4⋊C8—61no—
115#323—62no—
116C24×C3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4M1612yescomplemented
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—81no—
127C22×C3×C4—81no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132#235—21no—
133#235—21no—
134#235—21no—
135#235—21no—
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—42no—
144C2×C3×C8—41no—
145C2×C3×C8—81no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—21no—
155#214—24no—
156#213—24no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159C4⋊Dic3C22⋊C411yescharacteristic
160C4⋊Dic3C22⋊C411yescharacteristic
161Dic3⋊C4—21no—
162Dic3⋊C4—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166#208—21no—
167#208—21no—
168#208—21no—
169#208M1612yescomplemented
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174ΓL(1,𝔽25)—44no—
175C2×#41—31no—
176C22×D4C4×D311yescharacteristic
177C23×C4—21no—
178AGL(1,ℤ/8)—62no—
179AGL(1,ℤ/8)—62no—
180C2×SD16—61no—
181C2×D8—61no—
182C2×M16—31no—
183C2×M16—31no—
184C2×M16—62no—
185C22×C8—21no—
186C22×C8—21no—
187C4×Q8—31no—
188C4×Q8—31no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—31no—
194C4×D4—31no—
195C4×D4—61no—
196C4×D4—61no—
197#117—31no—
198#117—61no—
199C2×C4⋊C4C4×D311yescharacteristic
200C2×C22⋊C4—21no—
201C2×C42—61no—
202C2×C42C22×D311yescharacteristic
203#107—61no—
204#106—61no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—62no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210#103—61no—
211#102—61no—
212#102—61no—
213#102—61no—
214C22⋊C8—32no—
215C22⋊C8—62no—
216C22⋊C8—61no—
217#97—62no—
218#97—61no—
219C4×C8—21no—
220C4×C8—61no—
221C4×C8—61no—
222C4×C8—61no—
223C4×C8#6711yescharacteristic
224C4×C8#6711yescharacteristic
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—81no—
229C23×C3—81no—
230C23×C3C4×D411yescharacteristic
231C22×D3—21no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—41no—
235C3×D4—41no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—82no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12311yescharacteristic
250C2×C3×C4C2×M1611yescharacteristic
251C2×C3×C4C2×M1611yescharacteristic
252C2×C3×C4#13111yescharacteristic
253C2×C3×C4#13111yescharacteristic
254#67—41no—
255#67—41no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261C4×D3—42no—
262C4×D3—41no—
263C4×D3—41no—
264C4×D3—41no—
265C4×D3—41no—
266Dic6—21no—
267Dic6—21no—
268C3×C8—22no—
269C3×C8—41no—
270C3×C8—81no—
271C3⋊C8—22no—
272C3⋊C8—42no—
273C3⋊C8—41no—
274C3⋊C8—81no—
275C24—21no—
276#41—61no—
277#41—61no—
278#41—61no—
279#41—61no—
280C2×Q8—31no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—31no—
287C2×D4—61no—
288C2×D4ΓL(1,𝔽25)12yescomplemented
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4#23711yescharacteristic
300SD16—64no—
301D8—64no—
302M16—32no—
303M16—61no—
304M16—61no—
305M16—61no—
306M16—124no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—62no—
314C2×C8—81no—
315C2×C8—121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C42—61no—
328C42—61no—
329C42—61no—
330C42#23511yescharacteristic
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—42no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3#38611yescharacteristic
342C22×C3#43511yescharacteristic
343C22×C3#43611yescharacteristic
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351C3×C4#40711yescharacteristic
352C3×C4#40811yescharacteristic
353C3×C4#43511yescharacteristic
354C3×C4#43611yescharacteristic
355C3×C4#47111yescharacteristic
356C3×C4#47211yescharacteristic
357Dic3—41no—
358Dic3—41no—
359Dic3—41no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—61no—
364C23—81no—
365C23—81no—
366C23#67⋊C411yescharacteristic
367Q8—61no—
368Q8—61no—
369D4—22no—
370D4—22no—
371D4—41no—
372D4—41no—
373D4—61no—
374D4—61no—
375D4—121no—
376D4—121no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4D3×C22⋊C411yescharacteristic
394C2×C4#88311yescharacteristic
395C2×C4#88411yescharacteristic
396C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
397C2×C4#90111yescharacteristic
398C2×C4D3×M1611yescharacteristic
399C2×C4#90711yescharacteristic
400C2×C4C4×#6711yescharacteristic
401C2×C4#93711yescharacteristic
402C2×C4#93811yescharacteristic
403C8—22no—
404C8—41no—
405C8—62no—
406C8—81no—
407C8—122no—
408C8—121no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—82no—
413C2×C3#187011yescharacteristic
414C2×C3#190611yescharacteristic
415C2×C3#190711yescharacteristic
416D3—81no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—82no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22#472911yescharacteristic
428C22#473011yescharacteristic
429C22#495611yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—81no—
434C4—121no—
435C4—121no—
436C4—121no—
437C4#472611yescharacteristic
438C4#472711yescharacteristic
439C4#511111yescharacteristic
440C4#512811yescharacteristic
441C4#516011yescharacteristic
442C4#517711yescharacteristic
443C3#909711yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—82no—
447C2—241no—
448C2#6995811yescharacteristic
449C2#7021811yescharacteristic
450C2#7039111yescharacteristic
subgroupG/H
centerC2×C4#938
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC22×C3#386
prime radicalC2×#932C4
cosocleC2×C3×C42C23
hypercenterC2×D4⋊C8D3
hypocenterC3#9097
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #78447C2semidirect
3. C2×C3×D4⋊C8C2semidirect
4. #71601C2semidirect
5. #70425C2semidirect
6. #67706C2semidirect
7. #67656C2semidirect
37. C2×D4⋊C8D3semidirect
122. C2×C3×D4M16semidirect
169. #208M16semidirect
288. C2×D4ΓL(1,𝔽25)semidirect
443. C3#9097semidirect

Maximal subgroups

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

subgroupindex
C4×#9322
#784472
C2×C3×D4⋊C82
#716012
#704252
#677062
#676562
#90973