Finite Group Databasebeta

← #10690034

Subgroups — #10690034

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
1C3×#3101C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5C4×#832C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#8989—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C22×#208—21no—
12C3×#483—21no—
13C3×#476C411yescharacteristic, complemented
14C3×#476C411yescharacteristic, complemented
15C3×#461C411yescharacteristic, complemented
16C3×#461C411yescharacteristic, complemented
17C3×#450—21no—
18C3×C4×SD16—21no—
19C3×C4×SD16—21no—
20C3×#431C2211yescharacteristic
21C2×#832—21no—
22#5207—21no—
23#5012—21no—
24#5011—22no—
25D4×C3⋊C8—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4924C2211yescharacteristic
28C4×#208C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4535—22no—
31#4532—21no—
32#4532—21no—
33#4529—21no—
34#3101D311yescharacteristic
35D4×M16—31no—
36#1986—31no—
37C4×#102—31no—
38#1493—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—21no—
44C2×C3×SD16—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4C2×C411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—22no—
65C2×#208—21no—
66C2×#208—42no—
67C2×#208—42no—
68C2×#208D411yescharacteristic
69C2×#208D411yescharacteristic
70C22×C3⋊C8—22no—
71C22×C3⋊C8—21no—
72C3×#106—21no—
73C3×#106—21no—
74C3×C4⋊C8C2×C411yescharacteristic
75C3×C4⋊C8C2×C411yescharacteristic
76C3×#103—21no—
77C3×#103—21no—
78C3×#102—21no—
79C3×#102—21no—
80C3×C22⋊C8—21no—
81C3×C4×C8—21no—
82#832—24no—
83C3⋊#95—21no—
84#830—22no—
85#830—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C22×M16—61no—
93#533C2×D311yescharacteristic
94C2×C4×D4C2×D311yescharacteristic
95#483—21no—
96#476Dic311yescharacteristic, complemented
97#476Dic311yescharacteristic, complemented
98#461Dic311yescharacteristic, complemented
99#461Dic311yescharacteristic, complemented
100#450—21no—
101C4×SD16—21no—
102C4×SD16—21no—
103#436—61no—
104#435—62no—
105C8×D4—61no—
106#431C2×D311yescharacteristic
107#423—31no—
108C2×#102—61no—
109#407—61no—
110C4×M16—31no—
111C2×C4×C8—31no—
112C4×C4⋊C4—31no—
113#340—61no—
114#339—61no—
115#339—61no—
116Q8⋊C8—62no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×D4—22no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C2×C3×D4C22⋊C411yescharacteristic
124C2×C3×D4C22⋊C411yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—41no—
138C3×SD16—42no—
139C3×SD16—42no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C4⋊C4C22⋊C411yescharacteristic
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—41no—
156C3×C42—41no—
157C3×C42C22×C411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161C4⋊Dic3—22no—
162C4⋊Dic3—41no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165#208—21no—
166#208—21no—
167#208—21no—
168#208—44no—
169#208—41no—
170#208—41no—
171#208SD1612yescomplemented
172C2×C3⋊C8—21no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—44no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C22×D4#6711yescharacteristic
180C23×C4—21no—
181C2×SD16—21no—
182C2×SD16—21no—
183C2×M16—31no—
184C2×M16—31no—
185C2×M16—62no—
186C2×M16—61no—
187C2×M16—122no—
188C2×M16—122no—
189C22×C8—62no—
190C22×C8—61no—
191C4⋊Q8C2×Dic311yescharacteristic
192C4⋊Q8C2×Dic311yescharacteristic
193#125—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C2×Dic311yescharacteristic
201C4×D4C2×Dic311yescharacteristic
202C4×D4#6711yescharacteristic
203C4×D4#6711yescharacteristic
204#117—21no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C22⋊C4—21no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210#106—21no—
211#106—21no—
212C4⋊C8—32no—
213C4⋊C8—61no—
214C4⋊C8C2×Dic311yescharacteristic
215C4⋊C8C2×Dic311yescharacteristic
216#103—21no—
217#103—21no—
218#102—21no—
219#102—21no—
220#102—64no—
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—61no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3#12111yescharacteristic
237C3×Q8—42no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—21no—
243C3×D4—42no—
244C3×D4#10412yes—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
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—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C4×D411yescharacteristic
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12311yescharacteristic
268C2×C3×C4C2×SD1611yescharacteristic
269C2×C3×C4#13511yescharacteristic
270C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
271C2×C3×C4#13711yescharacteristic
272C2×Dic3—42no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C3×C8—41no—
276C3×C8—41no—
277C3⋊C8—22no—
278C3⋊C8—42no—
279C3⋊C8—42no—
280C3⋊C8—41no—
281C24—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4#21711yescharacteristic
288C2×D4#21711yescharacteristic
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—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—42no—
303SD16—42no—
304M16—32no—
305M16—61no—
306M16—61no—
307M16—61no—
308M16—124no—
309M16—121no—
310M16—121no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—61no—
316C2×C8—62no—
317C2×C8—124no—
318C2×C8—121no—
319C2×C8—121no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—62no—
328C4⋊C4—121no—
329C4⋊C4#21711yescharacteristic
330C4⋊C4#21711yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—41no—
336C42—41no—
337C42—121no—
338C42—121no—
339C42D3×D411yescharacteristic
340C42#23711yescharacteristic
341C42C22×Dic311yescharacteristic
342C42C2×#6711yescharacteristic
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—81no—
362C3×C4—81no—
363C3×C4C2×#10411yescharacteristic
364C3×C4#42111yescharacteristic
365C3×C4C4×SD1611yescharacteristic
366C3×C4#44011yescharacteristic
367C3×C4#45011yescharacteristic
368C3×C4#45211yescharacteristic
369Dic3—82no—
370C23—21no—
371C23—21no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23#93911yescharacteristic
377Q8—42no—
378Q8—41no—
379Q8—41no—
380D4—21no—
381D4—21no—
382D4—21no—
383D4—42no—
384D4#83712yes—
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—41no—
401C2×C4—41no—
402C2×C4—122no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4D3×SD1611yescharacteristic
406C2×C4#91411yescharacteristic
407C2×C4#91511yescharacteristic
408C2×C4#91611yescharacteristic
409C2×C4D4×Dic311yescharacteristic
410C2×C4D4×Dic311yescharacteristic
411C2×C4#93511yescharacteristic
412C2×C4#93711yescharacteristic
413C2×C4#93911yescharacteristic
414C2×C4C2×#21711yescharacteristic
415C8—41no—
416C8—41no—
417C8—62no—
418C8—122no—
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#516211yescharacteristic
439C22#516511yescharacteristic
440C22#522011yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—242no—
449C4Dic3×SD1611yescharacteristic
450C4#516411yescharacteristic
451C4#516911yescharacteristic
452C4#517311yescharacteristic
453C4C2×#83711yescharacteristic
454C4#523611yescharacteristic
455C3#898911yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#7171311yescharacteristic
461C2#7204711yescharacteristic
462C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3101D3
hypocenterC3#8989
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
3. #71841C2semidirect
4. #71826C2semidirect
6. #67746C2semidirect
7. #67739C2semidirect
13. C3×#476C4semidirect
14. C3×#476C4semidirect
15. C3×#461C4semidirect
16. C3×#461C4semidirect
96. #476Dic3semidirect
97. #476Dic3semidirect
98. #461Dic3semidirect
99. #461Dic3semidirect
171. #208SD16semidirect
455. C3#8989semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
D4×#2082
#718412
#718262
C4×#8322
#677462
#677392
#89893