Finite Group Databasebeta

← #10690051

Subgroups — #10690051

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
1#82442C211yesmaximal, characteristic
2Q8×#208C211yesmaximal, characteristic, complemented
3C3×C4×#103C211yesmaximal, characteristic, complemented
4#69735C211yesmaximal, characteristic
5#69492C211yesmaximal, characteristic
6#67656C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#8992—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C3×#103—21no—
12C2×C3×C4×C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5227—21no—
15#5027C411yescharacteristic, complemented
16#5027C411yescharacteristic, complemented
17C4×#215—21no—
18C4×#215—21no—
19#5024—21no—
20#5024—22no—
21Q8×C3⋊C8—21no—
22#5020C411yescharacteristic, complemented
23#5020C411yescharacteristic, complemented
24#4968—21no—
25C2×#805—21no—
26C2×#805—21no—
27#4925C2211yescharacteristic
28#4924—21no—
29#4924C2211yescharacteristic
30C4×#208—21no—
31C4×#208C2211yescharacteristic
32C3×#340—21no—
33#4459—22no—
34#3103—31no—
35Q8×M16—31no—
36#1980—31no—
37C4×#103D311yescharacteristic, complemented
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×C3×C8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8C2×C411yescharacteristic
47C3×C4×Q8C2×C411yescharacteristic
48C3×C4×Q8D411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—21no—
57C2×#215—21no—
58#930—21no—
59#930—21no—
60C2×#208—21no—
61C2×#208D411yescharacteristic
62C2×#208D411yescharacteristic
63#875—21no—
64#874C2×C411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic
68C3×#103—24no—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71C3×#95—21no—
72#835—21no—
73#835—21no—
74#830—21no—
75#809—21no—
76#809—21no—
77#808—21no—
78#808—21no—
79#805—42no—
80#805—42no—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—22no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)D412yes—
87#803—22no—
88#803—21no—
89C4×C3⋊C8—22no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92#532—31no—
93C2×C4×Q8C2×D311yescharacteristic
94#481—61no—
95#475—31no—
96#475—31no—
97#460—31no—
98#460—31no—
99#448—61no—
100#446—62no—
101#446—61no—
102C8×Q8—61no—
103C4×SD16—61no—
104C4×SD16—61no—
105#431—31no—
106#423—31no—
107#423—61no—
108C2×#104—61no—
109C2×#104—61no—
110C2×#103—21no—
111C4×M16—31no—
112C4×M16—61no—
113C2×C4×C8C2×D311yescharacteristic
114C4×C4⋊C4C2×D311yescharacteristic
115#340—21no—
116D4⋊C8—62no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—41no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×Q8C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42—41no—
155C3×C42—41no—
156C3×C42C22×C411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#215—42no—
161#215—42no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166#208—21no—
167#208—21no—
168#208—21no—
169#208—41no—
170#208—41no—
171#208SD1612yescomplemented
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177C2×#41—61no—
178C22×Q8#6711yescharacteristic
179C2×SD16—61no—
180C2×SD16—61no—
181C2×M16—31no—
182C2×M16—31no—
183C2×M16—61no—
184C22×C8—21no—
185C4⋊Q8—31no—
186#127—31no—
187#124—61no—
188#121—61no—
189#121—61no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8C4×D311yescharacteristic
193C4×Q8C4×D311yescharacteristic
194C4×Q8D1211yescharacteristic
195C4×Q8D1211yescharacteristic
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22×D311yescharacteristic
204#106—61no—
205#106—61no—
206C4⋊C8—31no—
207C4⋊C8—31no—
208C4⋊C8—32no—
209C4⋊C8—62no—
210C4⋊C8—62no—
211C4⋊C8—61no—
212#104—122no—
213#104—122no—
214#103—24no—
215#103—61no—
216#103—61no—
217#102—61no—
218#102—61no—
219C22⋊C8—61no—
220#97—62no—
221#97—61no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—62no—
225C4×C8—61no—
226C4×C8—61no—
227#95—21no—
228C23×C3#12111yescharacteristic
229C22×D3—41no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—42no—
234C3×Q8#10412yescomplemented
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12311yescharacteristic
253C2×C3×C4C2×SD1611yescharacteristic
254C2×C3×C4#13511yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256C2×C3×C4#13711yescharacteristic
257#67—81no—
258#67—81no—
259C2×Dic3—41no—
260D12—42no—
261D12—41no—
262D12—41no—
263D12—41no—
264D12—41no—
265C4×D3—81no—
266C4×D3—81no—
267Dic6—41no—
268Dic6—41no—
269C3×C8—42no—
270C3⋊C8—22no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C3⋊C8—41no—
274C3⋊C8—42no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—61no—
283C2×Q8(C2×D3)⋊C411yescharacteristic
284C2×Q8(C2×D3)⋊C411yescharacteristic
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—121no—
293C22×C4D3×D411yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—122no—
297SD16—122no—
298M16—32no—
299M16—61no—
300M16—61no—
301M16—61no—
302M16—121no—
303M16—121no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—61no—
308C2×C8—62no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4(C2×D3)⋊C411yescharacteristic
322C4⋊C4(C2×D3)⋊C411yescharacteristic
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—41no—
330C42—41no—
331C42C2×C4×D311yescharacteristic
332C42C2×D1211yescharacteristic
333C42D3×D411yescharacteristic
334C42#23711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3#38611yescharacteristic
338C22×C3#46011yescharacteristic
339C22×C3#46411yescharacteristic
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—82no—
351C3×C4C2×#10411yescharacteristic
352C3×C4#42111yescharacteristic
353C3×C4C4×SD1611yescharacteristic
354C3×C4#44011yescharacteristic
355C3×C4#44811yescharacteristic
356C3×C4#44911yescharacteristic
357Dic3—81no—
358C23—121no—
359C23#93911yescharacteristic
360Q8—21no—
361Q8—21no—
362Q8—21no—
363Q8—42no—
364Q8—121no—
365Q8—121no—
366Q8#80512yescomplemented
367D4—122no—
368D4—121no—
369D4—121no—
370D4—121no—
371D4—121no—
372D4—241no—
373D4—241no—
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—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—121no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4#67⋊C411yescharacteristic
390C2×C4#67⋊C411yescharacteristic
391C2×C4#88211yescharacteristic
392C2×C4#88611yescharacteristic
393C2×C4D3×SD1611yescharacteristic
394C2×C4#91411yescharacteristic
395C2×C4#91811yescharacteristic
396C2×C4#91911yescharacteristic
397C2×C4C2×(C2×D3)⋊C411yescharacteristic
398C2×C4#93911yescharacteristic
399C8—42no—
400C8—62no—
401C8—121no—
402C8—121no—
403C8—121no—
404C8—122no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3#187111yescharacteristic
408C2×C3#188011yescharacteristic
409C2×C3#188111yescharacteristic
410D3—161no—
411C22—21no—
412C22—21no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22#495811yescharacteristic
417C22#516311yescharacteristic
418C22#518511yescharacteristic
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4—241no—
428C4#478811yescharacteristic
429C4#480611yescharacteristic
430C4#481011yescharacteristic
431C4#481711yescharacteristic
432C4C2×#80511yescharacteristic
433C4#500511yescharacteristic
434C3#899211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—481no—
438C2#6945311yescharacteristic
439C2#7024011yescharacteristic
440C2#7029311yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#103C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC4×#103D3
hypocenterC3#8992
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. Q8×#208C2semidirect
3. C3×C4×#103C2semidirect
6. #67656C2semidirect
7. #67620C2semidirect
15. #5027C4semidirect
16. #5027C4semidirect
22. #5020C4semidirect
23. #5020C4semidirect
37. C4×#103D3semidirect
171. #208SD16semidirect
234. C3×Q8#104semidirect
366. Q8#805semidirect
434. C3#8992semidirect

Maximal subgroups

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

subgroupindex
#824422
Q8×#2082
C3×C4×#1032
#697352
#694922
#676562
#676202
#89923