Finite Group Databasebeta

← #10690007

Subgroups — #10690007

Order 768.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5083C211yesmaximal, characteristic
2C4⋊C4×D12C211yesmaximal, characteristic
3C3×#1988C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6#67702C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#8985—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×D3×C4⋊C4—21no—
12C2×C4×D12—21no—
13C2×C4×D12C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×#835—21no—
17#5083—22no—
18#5083C414yescomplemented
19#4987—21no—
20#4935C2211yescharacteristic
21#4928—21no—
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4668—22no—
25C3×#340—21no—
26C3×#340—21no—
27#4529—22no—
28#4483—22no—
29C2×#461—31no—
30D4×C4⋊C4—31no—
31#1990—31no—
32#1988D311yescharacteristic, complemented
33C2×D4⋊C8—31no—
34#1296—31no—
35#1279—31no—
36C22×C3×Q8—21no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C4×D3—22no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yescomplemented
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#215—24no—
52C2×#215—42no—
53C2×#215—42no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D412yes—
60C2×Dic3⋊C4—22no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63D3×C4⋊C4—42no—
64D3×C4⋊C4—42no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—42no—
68C4×D12—42no—
69C4×D12C2×C412yes—
70C4×D12D412yes—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#95—21no—
74#835—24no—
75C3⋊#95—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yescomplemented
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C22×SD16—61no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C4×D4—31no—
83C2×C4×D4—61no—
84C22×C4⋊C4—61no—
85#461—34no—
86#461—62no—
87C2×C4⋊C8—31no—
88C2×C4⋊C8C2×D311yescharacteristic
89C2×#103—61no—
90C2×C4×C8—31no—
91#389—61no—
92#385—62no—
93#384—31no—
94#380—61no—
95C4×C4⋊C4C2×D311yescharacteristic
96#340—21no—
97#340—21no—
98#340—62no—
99D4⋊C8—62no—
100C23×D3—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12C22⋊C412yes—
119C2×D12D812yes—
120C2×D12SD1612yes—
121C2×C4×D3—22no—
122C2×C4×D3—22no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—44no—
128C2×C4×D3—44no—
129C2×C4×D3—82no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4SD1612yescomplemented
141C3×C42—41no—
142C3×C42—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147#215—48no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—44no—
151C4⋊Dic3—21no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C4⋊Dic3—42no—
157C4⋊Dic3C22⋊C412yes—
158Dic3⋊C4—44no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×Q8—21no—
166C22×D4—31no—
167C23×C4—62no—
168C23×C4—61no—
169C2×SD16—64no—
170C2×SD16—122no—
171C2×SD16—122no—
172C22×C8—21no—
173C22×C8—61no—
174C22×C8—61no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C4×D312yescomplemented
179C4×D4—32no—
180C4×D4—32no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—122no—
184C4×D4—122no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4—122no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C4⋊C4#6711yescharacteristic
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C42—21no—
199C2×C42C22×D311yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—32no—
203C4⋊C8—61no—
204#103—64no—
205C4×C8—61no—
206C4×C8—61no—
207#95—21no—
208#95—62no—
209C23×C3#12011yescharacteristic
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—41no—
217C3×Q8—42no—
218C3×Q8—42no—
219C3×Q8—81no—
220C3×Q8—81no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4C2×SD1611yescharacteristic
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×C3×C4#13711yescharacteristic
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252D12—22no—
253D12—22no—
254D12—22no—
255D12#10214yes—
256C4×D3—42no—
257C4×D3—42no—
258C4×D3—42no—
259C4×D3—42no—
260C4×D3—88no—
261C3×C8—81no—
262C3⋊C8—42no—
263C3⋊C8—42no—
264C24—61no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×D4—32no—
272C2×D4—32no—
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—124no—
285C22×C4—124no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—121no—
290C22×C4—242no—
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4C2×#6711yescharacteristic
294SD16—128no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—32no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—61no—
313C4⋊C4—122no—
314C4⋊C4—124no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4#21512yescomplemented
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—124no—
323C42—41no—
324C42—41no—
325C42C2×C4×D311yescharacteristic
326C42#23511yescharacteristic
327C42D3×D411yescharacteristic
328C42#23711yescharacteristic
329C22×C3#38611yescharacteristic
330C22×C3#44711yescharacteristic
331C22×C3#44811yescharacteristic
332C22×C3#45011yescharacteristic
333C22×C3#45211yescharacteristic
334C22×C3#45511yescharacteristic
335C22×C3#45611yescharacteristic
336C2×D3—24no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—42no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—82no—
347C3×C4—81no—
348C3×C4C2×#10211yescharacteristic
349C3×C4#41711yescharacteristic
350C3×C4C4×SD1611yescharacteristic
351C3×C4#44011yescharacteristic
352C3×C4#46111yescharacteristic
353C3×C4#46311yescharacteristic
354C3×C4#48011yescharacteristic
355C3×C4#48111yescharacteristic
356Dic3—42no—
357Dic3—82no—
358C23—62no—
359C23—62no—
360C23—62no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23#93711yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—34no—
371D4—62no—
372D4—62no—
373D4—62no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—62no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—248no—
396C2×C4D3×C22⋊C411yescharacteristic
397C2×C4#67⋊C411yescharacteristic
398C2×C4#88311yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4D3×D811yescharacteristic
401C2×C4#91111yescharacteristic
402C2×C4D3×SD1611yescharacteristic
403C2×C4#91511yescharacteristic
404C2×C4C4×#6711yescharacteristic
405C2×C4#93811yescharacteristic
406C2×C4C2×#21511yescharacteristic
407C2×C4#94211yescharacteristic
408C2×C4#94911yescharacteristic
409C2×C4#95011yescharacteristic
410C8—81no—
411C8—122no—
412C8—122no—
413C2×C3#163911yescharacteristic
414C2×C3#164311yescharacteristic
415C2×C3#164711yescharacteristic
416C2×C3#165111yescharacteristic
417C2×C3#187411yescharacteristic
418C2×C3#187711yescharacteristic
419C2×C3#188211yescharacteristic
420D3—44no—
421C22—64no—
422C22—122no—
423C22—122no—
424C22—122no—
425C22—122no—
426C22—122no—
427C22—122no—
428C22#495611yescharacteristic
429C22#504611yescharacteristic
430C22#504711yescharacteristic
431C22#515611yescharacteristic
432C22#517011yescharacteristic
433C22#520211yescharacteristic
434C22#520511yescharacteristic
435C4—21no—
436C4—21no—
437C4—42no—
438C4—82no—
439C4—81no—
440C4—122no—
441C4—242no—
442C4D3×#10211yescharacteristic
443C4#477111yescharacteristic
444C4#477411yescharacteristic
445C4#477711yescharacteristic
446C4C4×#21511yescharacteristic
447C4#502611yescharacteristic
448C4#505711yescharacteristic
449C4#508311yescharacteristic
450C3#898511yescharacteristic, complemented, Sylow
451C2—124no—
452C2#6968711yescharacteristic
453C2#7020011yescharacteristic
454C2#7034811yescharacteristic
455C2#7110811yescharacteristic
456C2#7111011yescharacteristic
457C2#7112511yescharacteristic
458C2#7112711yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1988D3
hypocenterC3#8985
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. C3×#1988C2semidirect
4. #71787C2semidirect
6. #67702C2semidirect
7. #67619C2semidirect
18. #5083C4semidirect
32. #1988D3semidirect
44. C3×C4⋊Q8C2×C4semidirect
77. C4⋊(C3⋊C8)C2×C4semidirect
140. C3×C4⋊C4SD16semidirect
178. C4⋊Q8C4×D3semidirect
319. C4⋊C4#215semidirect
450. C3#8985semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
C4⋊C4×D122
C3×#19882
#717872
C2×#44832
#677022
#676192
#89853