Finite Group Databasebeta

← #10690028

Subgroups — #10690028

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5083C211yesmaximal, characteristic
2#73638C211yesmaximal, characteristic
3C3×#1988C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6#67702C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#8988—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#892—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#835—21no—
16#5083—22no—
17#5083C414yescomplemented
18#4987—21no—
19#4939—21no—
20#4935C2211yescharacteristic
21C4×C4⋊Dic3—21no—
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4670—22no—
25C3×#340—21no—
26C3×#340—21no—
27#4529—22no—
28#4483—22no—
29C2×#461—31no—
30#2354—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×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51C2×#215—42no—
52C2×#215—42no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D412yes—
60C2×C4×Dic3—22no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63#892—42no—
64#892—42no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12C2×C412yes—
68C4×D12D412yes—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#95—21no—
72#835—24no—
73C3⋊#95—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)C2×C412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C22×SD16—61no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4×D4—31no—
81C2×#117—61no—
82#461—34no—
83#461—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#103—61no—
87C2×C4×C8—31no—
88#390—61no—
89#388—62no—
90#384—31no—
91#380—61no—
92C4×C4⋊C4—61no—
93C4×C4⋊C4C2×D311yescharacteristic
94#340—21no—
95#340—21no—
96#340—62no—
97D4⋊C8—62no—
98C23×D3—21no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12C22⋊C412yes—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4SD1612yescomplemented
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142#215—48no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—82no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3—42no—
153C4⋊Dic3C22⋊C412yes—
154C4×Dic3—44no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—64no—
165C2×SD16—122no—
166C2×SD16—122no—
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C4×D312yes—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—61no—
177C4×D4—61no—
178#117—122no—
179#117—122no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C42—21no—
192C2×C42—62no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198#103—64no—
199C4×C8—61no—
200C4×C8—61no—
201#95—21no—
202#95—62no—
203C23×C3#12011yescharacteristic
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4#13711yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246D12—22no—
247D12—22no—
248D12—41no—
249D12—41no—
250C4×D3—42no—
251C4×D3—42no—
252C4×D3—42no—
253C4×D3—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257C24—61no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—32no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284SD16—128no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—32no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4#21512yescomplemented
309C22⋊C4—62no—
310C22⋊C4—62no—
311C22⋊C4—124no—
312C22⋊C4—242no—
313C42—41no—
314C42—41no—
315C42—124no—
316C42C2×C4×D311yescharacteristic
317C42#23511yescharacteristic
318C42D3×D411yescharacteristic
319C42#23711yescharacteristic
320C22×C3#38611yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#44911yescharacteristic
323C22×C3#45011yescharacteristic
324C22×C3#45211yescharacteristic
325C22×C3#45511yescharacteristic
326C22×C3#45611yescharacteristic
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—82no—
338C3×C4—81no—
339C3×C4#41611yescharacteristic
340C3×C4#41811yescharacteristic
341C3×C4C4×SD1611yescharacteristic
342C3×C4#44011yescharacteristic
343C3×C4#46111yescharacteristic
344C3×C4#46311yescharacteristic
345C3×C4#48211yescharacteristic
346C3×C4#48211yescharacteristic
347Dic3—42no—
348Dic3—82no—
349C23—62no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23#93711yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—81no—
361D4—62no—
362D4—62no—
363D4—121no—
364D4—121no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—62no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4D3×C22⋊C411yescharacteristic
387C2×C4#67⋊C411yescharacteristic
388C2×C4#88311yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4#91111yescharacteristic
391C2×C4#91211yescharacteristic
392C2×C4#91411yescharacteristic
393C2×C4#91611yescharacteristic
394C2×C4C4×#6711yescharacteristic
395C2×C4#93811yescharacteristic
396C2×C4C2×#21511yescharacteristic
397C2×C4#94211yescharacteristic
398C2×C4#94911yescharacteristic
399C2×C4#95011yescharacteristic
400C8—81no—
401C8—122no—
402C8—122no—
403C2×C3#164211yescharacteristic
404C2×C3#164611yescharacteristic
405C2×C3#165011yescharacteristic
406C2×C3#165411yescharacteristic
407C2×C3#187511yescharacteristic
408C2×C3#187711yescharacteristic
409C2×C3#188211yescharacteristic
410D3—82no—
411C22—122no—
412C22—122no—
413C22—122no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22#495611yescharacteristic
419C22#504611yescharacteristic
420C22#504711yescharacteristic
421C22#515811yescharacteristic
422C22#517211yescharacteristic
423C22#520211yescharacteristic
424C22#520511yescharacteristic
425C4—21no—
426C4—21no—
427C4—42no—
428C4—82no—
429C4—81no—
430C4—122no—
431C4—242no—
432C4#477011yescharacteristic
433C4#477211yescharacteristic
434C4#477911yescharacteristic
435C4#478411yescharacteristic
436C4C4×#21511yescharacteristic
437C4#502611yescharacteristic
438C4#505711yescharacteristic
439C4#508311yescharacteristic
440C3#898811yescharacteristic, complemented, Sylow
441C2—242no—
442C2#6968711yescharacteristic
443C2#7020111yescharacteristic
444C2#7034811yescharacteristic
445C2#7110911yescharacteristic
446C2#7111111yescharacteristic
447C2#7112411yescharacteristic
448C2#7112611yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1988D3
hypocenterC3#8988
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
17. #5083C4semidirect
32. #1988D3semidirect
135. C3×C4⋊C4SD16semidirect
308. C4⋊C4#215semidirect
440. C3#8988semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
#736382
C3×#19882
#717872
C2×#44832
#677022
#676192
#89883