Finite Group Databasebeta

← #10690030

Subgroups — #10690030

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#82314C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic
3C3×C4×#102C211yesmaximal, characteristic
4#69738C211yesmaximal, characteristic, complemented
5#69493C211yesmaximal, characteristic, complemented
6#67656C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#8989—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#208—21no—
11#5470C2211yescharacteristic
12C2×C3×#102—21no—
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5218—21no—
16#5207—21no—
17#5019C411yescharacteristic, complemented
18#5019C411yescharacteristic, complemented
19C4×#214—21no—
20C4×#214—21no—
21#5012—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#5008C411yescharacteristic, complemented
25#5008C411yescharacteristic, complemented
26#4971—21no—
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29C4×#208C2211yescharacteristic
30C3×#340—21no—
31#4523—21no—
32#4523—21no—
33#4456—22no—
34#3101—31no—
35D4×M16—31no—
36#1986—31no—
37C4×#102D311yescharacteristic
38#1493—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×C3×C8—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4C2×C411yescharacteristic
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4D411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#214—21no—
56C2×#214—21no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—22no—
61C2×#208—21no—
62C2×#208—42no—
63C2×#208—42no—
64C2×#208D411yescharacteristic
65C2×#208D411yescharacteristic
66C22×C3⋊C8—22no—
67#870—21no—
68#869C2×C411yescharacteristic
69#869C2×C411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#102—24no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3×#95—21no—
76#832—21no—
77#832—21no—
78#830—22no—
79#830—22no—
80#830—21no—
81#810—21no—
82#810—21no—
83#808—21no—
84#808—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)D412yes—
89#803—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C22×M16—61no—
93#533—31no—
94C2×C4×D4C2×D311yescharacteristic
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#431—31no—
107#423—31no—
108C2×#102—21no—
109#407—61no—
110C4×M16—31no—
111C2×C4×C8C2×D311yescharacteristic
112C4×C4⋊C4C2×D311yescharacteristic
113#340—21no—
114#339—61no—
115#339—61no—
116Q8⋊C8—62no—
117C24×C3—21no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4C22⋊C411yescharacteristic
122C2×C3×D4C22⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42C22×C411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#217—41no—
157#214—42no—
158#214—42no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166#208—21no—
167#208—21no—
168#208—21no—
169#208—44no—
170#208—41no—
171#208—41no—
172#208SD1612yescomplemented
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—44no—
179C22×D4#6711yescharacteristic
180C23×C4—21no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×M16—31no—
184C2×M16—31no—
185C2×M16—62no—
186C2×M16—61no—
187C2×M16—122no—
188C2×M16—122no—
189C22×C8—21no—
190C22×C8—62no—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193#125—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C4×D311yescharacteristic
201C4×D4C4×D311yescharacteristic
202C4×D4D1211yescharacteristic
203C4×D4D1211yescharacteristic
204#117—61no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42C22×D311yescharacteristic
210#106—61no—
211#106—61no—
212C4⋊C8—31no—
213C4⋊C8—31no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216#103—61no—
217#103—61no—
218#102—24no—
219#102—61no—
220#102—61no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8—61no—
224#97—61no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—61no—
228C4×C8—61no—
229#95—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3#12111yescharacteristic
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—42no—
241C3×D4#10412yes—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4C4×D411yescharacteristic
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12311yescharacteristic
266C2×C3×C4C2×SD1611yescharacteristic
267C2×C3×C4#13511yescharacteristic
268C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
269C2×C3×C4#13711yescharacteristic
270C2×Dic3—41no—
271C2×Dic3—41no—
272Dic6—42no—
273Dic6—41no—
274Dic6—41no—
275C3×C8—42no—
276C3⋊C8—22no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—42no—
280C3⋊C8—41no—
281C24—21no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4(C2×D3)⋊C411yescharacteristic
288C2×D4(C2×D3)⋊C411yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303SD16—122no—
304M16—32no—
305M16—61no—
306M16—61no—
307M16—61no—
308M16—124no—
309M16—121no—
310M16—121no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—62no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—124no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4(C2×D3)⋊C411yescharacteristic
330C4⋊C4(C2×D3)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—121no—
336C42—41no—
337C42—41no—
338C42—121no—
339C42C2×C4×D311yescharacteristic
340C42C2×D1211yescharacteristic
341C42D3×D411yescharacteristic
342C42#23711yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3#38611yescharacteristic
354C22×C3#46111yescharacteristic
355C22×C3#46311yescharacteristic
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—82no—
362C3×C4C2×#10411yescharacteristic
363C3×C4#42111yescharacteristic
364C3×C4C4×SD1611yescharacteristic
365C3×C4#44011yescharacteristic
366C3×C4#45011yescharacteristic
367C3×C4#45211yescharacteristic
368Dic3—81no—
369Dic3—81no—
370C23—21no—
371C23—21no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23#93911yescharacteristic
377Q8—122no—
378Q8—121no—
379Q8—121no—
380D4—21no—
381D4—21no—
382D4—21no—
383D4—42no—
384D4#80512yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4#67⋊C411yescharacteristic
406C2×C4#67⋊C411yescharacteristic
407C2×C4#88211yescharacteristic
408C2×C4#88611yescharacteristic
409C2×C4#91111yescharacteristic
410C2×C4#91211yescharacteristic
411C2×C4D3×SD1611yescharacteristic
412C2×C4#91511yescharacteristic
413C2×C4C2×(C2×D3)⋊C411yescharacteristic
414C2×C4#93911yescharacteristic
415C8—42no—
416C8—62no—
417C8—121no—
418C8—121no—
419C8—122no—
420C8—121no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—42no—
424C2×C3—81no—
425C2×C3#187111yescharacteristic
426C2×C3#187711yescharacteristic
427C2×C3#188111yescharacteristic
428C22—21no—
429C22—21no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—82no—
437C22—82no—
438C22#495811yescharacteristic
439C22#515311yescharacteristic
440C22#516211yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—241no—
448C4—241no—
449C4#475711yescharacteristic
450C4#475811yescharacteristic
451C4#477611yescharacteristic
452C4#478311yescharacteristic
453C4C2×#80511yescharacteristic
454C4#500511yescharacteristic
455C3#898911yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#6945311yescharacteristic
461C2#7017111yescharacteristic
462C2#7029311yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#102C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterC4×#102D3
hypocenterC3#8989
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
4. #69738C2semidirect
5. #69493C2semidirect
6. #67656C2semidirect
7. #67620C2semidirect
17. #5019C4semidirect
18. #5019C4semidirect
24. #5008C4semidirect
25. #5008C4semidirect
172. #208SD16semidirect
455. C3#8989semidirect

Maximal subgroups

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

subgroupindex
#823142
D4×#2082
C3×C4×#1022
#697382
#694932
#676562
#676202
#89893