Finite Group Databasebeta

← #10690055

Subgroups — #10690055

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3103C211yesmaximal, characteristic
2Q8×#208C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic
4#71802C211yesmaximal, characteristic
5C4×#835C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#8992—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#481—21no—
12C3×#475C411yescharacteristic, complemented
13C3×#475C411yescharacteristic, complemented
14C3×#460C411yescharacteristic, complemented
15C3×#460C411yescharacteristic, complemented
16C3×#448—21no—
17C3×C4×SD16—21no—
18C3×C4×SD16—21no—
19C3×#431C2211yescharacteristic
20C2×#837—21no—
21C2×#837—21no—
22C2×#835—21no—
23#5024—21no—
24#5024—22no—
25Q8×C3⋊C8—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4924—21no—
28#4924C2211yescharacteristic
29C4×#208—21no—
30C4×#208C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4529—21no—
34#3103D311yescharacteristic
35Q8×M16—31no—
36#1980—31no—
37C4×#103—31no—
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×SD16—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8C2×C411yescharacteristic
53C3×C4×Q8C2×C411yescharacteristic
54C3×C4×Q8D411yescharacteristic
55C3×C4×Q8D411yescharacteristic
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C2×#208D411yescharacteristic
66C2×#208D411yescharacteristic
67C22×C3⋊C8—21no—
68C3×#106—21no—
69C3×#106—21no—
70C3×C4⋊C8C2×C411yescharacteristic
71C3×C4⋊C8C2×C411yescharacteristic
72C3×#103—21no—
73C3×#103—21no—
74C3×#102—21no—
75C3×#102—21no—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78#837—42no—
79#837—42no—
80#835—24no—
81C3⋊#95—21no—
82C4⋊(C3⋊C8)—22no—
83C4⋊(C3⋊C8)—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86#803—22no—
87#803—21no—
88C4×C3⋊C8—22no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92#532C2×D311yescharacteristic
93C2×C4×Q8C2×D311yescharacteristic
94#481—21no—
95#475Dic311yescharacteristic, complemented
96#475Dic311yescharacteristic, complemented
97#460Dic311yescharacteristic, complemented
98#460Dic311yescharacteristic, complemented
99#448—21no—
100#446—62no—
101#446—61no—
102C8×Q8—61no—
103C4×SD16—21no—
104C4×SD16—21no—
105#431C2×D311yescharacteristic
106#423—31no—
107#423—61no—
108C2×#104—61no—
109C2×#104—61no—
110C2×#103—61no—
111C4×M16—31no—
112C4×M16—61no—
113C2×C4×C8—31no—
114C4×C4⋊C4—31no—
115#340—61no—
116D4⋊C8—62no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8—22no—
123C2×C3×Q8—21no—
124C2×C3×Q8—41no—
125C2×C3×Q8C22⋊C411yescharacteristic
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—41no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
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—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C4⋊C4C22⋊C411yescharacteristic
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42—21no—
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—41no—
169#208—41no—
170#208SD1612yescomplemented
171C2×C3⋊C8—21no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41—21no—
178C22×Q8#6711yescharacteristic
179C2×SD16—21no—
180C2×SD16—21no—
181C2×M16—31no—
182C2×M16—31no—
183C2×M16—61no—
184C22×C8—61no—
185C4⋊Q8C2×Dic311yescharacteristic
186#127C2×Dic311yescharacteristic
187#124—21no—
188#121—21no—
189#121—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8C2×Dic311yescharacteristic
193C4×Q8C2×Dic311yescharacteristic
194C4×Q8#6711yescharacteristic
195C4×Q8#6711yescharacteristic
196C4×D4—21no—
197C4×D4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C42—21no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#106—21no—
205#106—21no—
206C4⋊C8—32no—
207C4⋊C8—62no—
208C4⋊C8—62no—
209C4⋊C8—61no—
210C4⋊C8C2×Dic311yescharacteristic
211C4⋊C8C2×Dic311yescharacteristic
212#104—122no—
213#104—122no—
214#103—21no—
215#103—21no—
216#103—64no—
217#102—21no—
218#102—21no—
219C22⋊C8—21no—
220#97—62no—
221#97—61no—
222C4×C8—21no—
223C4×C8—62no—
224C4×C8—61no—
225C4×C8—61no—
226C4×C8—61no—
227#95—61no—
228C23×C3—41no—
229C23×C3#12111yescharacteristic
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—42no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8#10412yescomplemented
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—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12311yescharacteristic
262C2×C3×C4C2×SD1611yescharacteristic
263C2×C3×C4#13511yescharacteristic
264C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
265C2×C3×C4#13711yescharacteristic
266C2×Dic3—42no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269C3×C8—41no—
270C3×C8—41no—
271C3⋊C8—22no—
272C3⋊C8—42no—
273C3⋊C8—41no—
274C3⋊C8—42no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—41no—
283C2×Q8#21711yescharacteristic
284C2×Q8#21711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—121no—
293C22×C4D3×D411yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—42no—
297SD16—42no—
298M16—32no—
299M16—61no—
300M16—61no—
301M16—61no—
302M16—121no—
303M16—121no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—61no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—61no—
310C2×C8—121no—
311C2×C8—121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—62no—
320C4⋊C4—121no—
321C4⋊C4#21711yescharacteristic
322C4⋊C4#21711yescharacteristic
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—121no—
330C42—121no—
331C42D3×D411yescharacteristic
332C42#23711yescharacteristic
333C42C22×Dic311yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#38611yescharacteristic
341C22×C3#46011yescharacteristic
342C22×C3#46411yescharacteristic
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—81no—
351C3×C4C2×#10411yescharacteristic
352C3×C4#42111yescharacteristic
353C3×C4C4×SD1611yescharacteristic
354C3×C4#44011yescharacteristic
355C3×C4#44811yescharacteristic
356C3×C4#44911yescharacteristic
357Dic3—82no—
358C23—41no—
359C23#93911yescharacteristic
360Q8—21no—
361Q8—21no—
362Q8—21no—
363Q8—42no—
364Q8—41no—
365Q8—41no—
366Q8#83712yescomplemented
367D4—42no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—122no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4D3×SD1611yescharacteristic
390C2×C4#91411yescharacteristic
391C2×C4#91511yescharacteristic
392C2×C4#91611yescharacteristic
393C2×C4D4×Dic311yescharacteristic
394C2×C4D4×Dic311yescharacteristic
395C2×C4#93511yescharacteristic
396C2×C4#93711yescharacteristic
397C2×C4#93911yescharacteristic
398C2×C4C2×#21711yescharacteristic
399C8—41no—
400C8—41no—
401C8—62no—
402C8—122no—
403C8—121no—
404C8—122no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—161no—
408C2×C3#187111yescharacteristic
409C2×C3#188011yescharacteristic
410C2×C3#188111yescharacteristic
411C22—21no—
412C22—21no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#516311yescharacteristic
417C22#516611yescharacteristic
418C22#522011yescharacteristic
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—81no—
427C4—242no—
428C4Dic3×SD1611yescharacteristic
429C4#516411yescharacteristic
430C4#517011yescharacteristic
431C4#517211yescharacteristic
432C4C2×#83711yescharacteristic
433C4#523611yescharacteristic
434C3#899211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—161no—
438C2#7171311yescharacteristic
439C2#7204811yescharacteristic
440C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3103C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3103D3
hypocenterC3#8992
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. Q8×#208C2semidirect
5. C4×#835C2semidirect
6. #67746C2semidirect
7. #67739C2semidirect
12. C3×#475C4semidirect
13. C3×#475C4semidirect
14. C3×#460C4semidirect
15. C3×#460C4semidirect
95. #475Dic3semidirect
96. #475Dic3semidirect
97. #460Dic3semidirect
98. #460Dic3semidirect
170. #208SD16semidirect
236. C3×Q8#104semidirect
366. Q8#837semidirect
434. C3#8992semidirect

Maximal subgroups

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

subgroupindex
C3×#31032
Q8×#2082
#718182
#718022
C4×#8352
#677462
#677392
#89923