Finite Group Databasebeta

← #10690033

Subgroups — #10690033

Order 768.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79298C211yesmaximal, characteristic
2#79043C211yesmaximal, characteristic
3#70501C211yesmaximal, characteristic, complemented
4#69493C211yesmaximal, characteristic, complemented
5C4×#809C211yesmaximal, characteristic
6C3×#1303C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#8989—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5126—21no—
16C2×#809—21no—
17C4×#208C2211yescharacteristic
18C2×C4×C3⋊C8C2211yescharacteristic
19#4842C411yescharacteristic, complemented
20#4842C411yescharacteristic, complemented
21#4841C411yescharacteristic, complemented
22#4841C411yescharacteristic, complemented
23#4838—21no—
24#4836—22no—
25#4834—21no—
26#4731—21no—
27#4721—21no—
28C4×#204—21no—
29C4×#204—21no—
30#4529—21no—
31#4523—21no—
32#4523—21no—
33#4484—22no—
34#3101—31no—
35D4×M16—31no—
36#1986—31no—
37C4×#102—31no—
38#1493—31no—
39#1303D311yescharacteristic, complemented
40#1278—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48#926—21no—
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54C22×C3⋊C8—21no—
55D3×M16—42no—
56D3×M16—42no—
57C2×#204—21no—
58C2×#204—21no—
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12C2×C411yescharacteristic
64C4×D12C2×C411yescharacteristic
65C4×D12D411yescharacteristic
66C4×D12D411yescharacteristic
67#870—21no—
68#869C2×C411yescharacteristic
69#869C2×C411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8C2×C411yescharacteristic
74C3×C4⋊C8C2×C411yescharacteristic
75C3×C4⋊C8D412yes—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78C3×#95—21no—
79#821—21no—
80#821—21no—
81(C2×D3)⋊C8—22no—
82(C2×D3)⋊C8—22no—
83#817—21no—
84#817—21no—
85#816—21no—
86#816—21no—
87#809—24no—
88#803—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C22×M16—61no—
93#533—31no—
94C2×C4×D4—31no—
95#483—61no—
96#476—31no—
97#476—31no—
98#461—31no—
99#461—31no—
100#450—61no—
101C4×SD16—61no—
102C4×SD16—61no—
103#436—61no—
104#435—62no—
105C8×D4—61no—
106#431C2×D311yescharacteristic
107#423C2×D311yescharacteristic
108C2×#102—61no—
109#407—61no—
110C4×M16—31no—
111C2×C4×C8—31no—
112C4×C4⋊C4C2×D311yescharacteristic
113#340—61no—
114#339—61no—
115#339—61no—
116Q8⋊C8—62no—
117C23×D3—21no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C2×D12—22no—
124C2×D12—21no—
125C2×D12—41no—
126C2×D12C22⋊C411yescharacteristic
127C2×D12C22⋊C411yescharacteristic
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C22×C411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—41no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166#208—21no—
167#208—21no—
168#208—21no—
169#208SD1612yescomplemented
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175#204—42no—
176#204—42no—
177ΓL(1,𝔽25)—44no—
178D3×C8—44no—
179C22×D4—31no—
180C23×C4—61no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×M16—21no—
184C2×M16—31no—
185C2×M16—31no—
186C2×M16—62no—
187C2×M16—122no—
188C2×M16—122no—
189C22×C8—62no—
190C22×C8—61no—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193#125—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—31no—
201C4×D4—31no—
202C4×D4—61no—
203C4×D4—61no—
204#117—61no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C22⋊C4—61no—
208C2×C42—21no—
209C2×C42C22×D311yescharacteristic
210#106—61no—
211#106—61no—
212C4⋊C8—21no—
213C4⋊C8C4×D311yescharacteristic
214C4⋊C8C4×D311yescharacteristic
215C4⋊C8D1212yes—
216#103—61no—
217#103—61no—
218#102—64no—
219#102—61no—
220#102—61no—
221C22⋊C8—21no—
222C22⋊C8—62no—
223C22⋊C8—62no—
224#97—61no—
225C4×C8—21no—
226C4×C8—61no—
227C4×C8—61no—
228C4×C8—61no—
229#95—21no—
230C23×C3#12111yescharacteristic
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—81no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×C3×C4C2×SD1611yescharacteristic
250C2×C3×C4#13511yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4#13711yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259D12—21no—
260D12—21no—
261D12—21no—
262D12—42no—
263D12#10412yes—
264C4×D3—41no—
265C4×D3—41no—
266C4×D3—41no—
267C4×D3—41no—
268C4×D3—41no—
269C4×D3—41no—
270C4×D3—41no—
271C4×D3—41no—
272Dic6—42no—
273Dic6—41no—
274Dic6—41no—
275C3×C8—41no—
276C3×C8—41no—
277C3×C8—42no—
278C3⋊C8—22no—
279C3⋊C8—42no—
280C3⋊C8—41no—
281C24—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—62no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—41no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4#23511yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302SD16—122no—
303SD16—122no—
304M16—32no—
305M16—41no—
306M16—41no—
307M16—61no—
308M16—61no—
309M16—61no—
310M16—124no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—61no—
315C2×C8—62no—
316C2×C8—61no—
317C2×C8—124no—
318C2×C8—121no—
319C2×C8—121no—
320C4⋊C4—22no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—121no—
336C42—41no—
337C42—41no—
338C42—121no—
339C42C2×C4×D311yescharacteristic
340C42C2×D1211yescharacteristic
341C42D3×D411yescharacteristic
342C42#23911yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3#38611yescharacteristic
346C22×C3#46111yescharacteristic
347C22×C3#46311yescharacteristic
348C2×D3—22no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—82no—
360C3×C4C2×#10411yescharacteristic
361C3×C4#42111yescharacteristic
362C3×C4C4×SD1611yescharacteristic
363C3×C4#44011yescharacteristic
364C3×C4#45011yescharacteristic
365C3×C4#45211yescharacteristic
366Dic3—41no—
367Dic3—41no—
368Dic3—81no—
369Dic3—81no—
370C23—61no—
371C23—61no—
372C23—122no—
373C23—121no—
374C23—121no—
375C23—241no—
376C23#88411yescharacteristic
377Q8—122no—
378Q8—121no—
379Q8—121no—
380D4—32no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—122no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4C4×D1211yescharacteristic
406C2×C4D3×C22⋊C411yescharacteristic
407C2×C4#88211yescharacteristic
408C2×C4#89311yescharacteristic
409C2×C4#89411yescharacteristic
410C2×C4#89511yescharacteristic
411C2×C4#91111yescharacteristic
412C2×C4D3×SD1611yescharacteristic
413C2×C4#91611yescharacteristic
414C2×C4#91811yescharacteristic
415C8—42no—
416C8—41no—
417C8—41no—
418C8—62no—
419C8—122no—
420C8—121no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3#187111yescharacteristic
424C2×C3#187711yescharacteristic
425C2×C3#188111yescharacteristic
426D3—42no—
427D3—81no—
428C22—21no—
429C22—21no—
430C22—62no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22—242no—
437C22—242no—
438C22#466511yescharacteristic
439C22#478711yescharacteristic
440C22#481811yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—121no—
446C4—121no—
447C4—241no—
448C4—241no—
449C4#477611yescharacteristic
450C4#480811yescharacteristic
451C4D3×#10411yescharacteristic
452C4#482111yescharacteristic
453C4#485211yescharacteristic
454C4#488911yescharacteristic
455C3#898911yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—122no—
459C2—241no—
460C2#6875511yescharacteristic
461C2#6889611yescharacteristic
462C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#8989
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #70501C2semidirect
4. #69493C2semidirect
6. C3×#1303C2semidirect
7. #67739C2semidirect
19. #4842C4semidirect
20. #4842C4semidirect
21. #4841C4semidirect
22. #4841C4semidirect
39. #1303D3semidirect
169. #208SD16semidirect
455. C3#8989semidirect

Maximal subgroups

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

subgroupindex
#792982
#790432
#705012
#694932
C4×#8092
C3×#13032
#677392
#89893