Finite Group Databasebeta

← #10691374

Subgroups — #10691374

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4848C211yesmaximal, characteristic
2C2×C8×Dic6C211yesmaximal, characteristic, complemented
3#70485C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic
5#69772C211yesmaximal, characteristic
6C3×#1277C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9222—31nomaximal, Sylow
9C22×#204—21no—
10C2×#874C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#821—21no—
16C2×#821—21no—
17C2×#821—21no—
18C2×#816—21no—
19C2×Dic3⋊C8—21no—
20C2×C8×Dic3—21no—
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23#4848—22no—
24#4848C414yescomplemented
25C8×Dic6—22no—
26C8×Dic6—22no—
27#4551—21no—
28#4485—22no—
29C2×#460—31no—
30C2×C8×Q8—31no—
31#1957—31no—
32#1803—31no—
33C2×#331—31no—
34#1296—31no—
35#1277D311yescharacteristic, complemented
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×Dic6D411yescharacteristic
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C22×C3×C8D411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C2×#204—24no—
53C2×#204—42no—
54C2×#204—42no—
55#874—21no—
56#874—21no—
57#874—21no—
58#874C2×C412yes—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C412yes—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69#821—24no—
70#821—42no—
71#821—42no—
72#816—42no—
73Dic3⋊C8—24no—
74C8×Dic3—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C22×SD16—61no—
78C2×#127—31no—
79C2×C4×Q8—31no—
80#460—34no—
81#460—62no—
82C8×Q8—62no—
83C8×Q8—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8—61no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—61no—
89C2×#102—61no—
90C2×#102—61no—
91C2×#102—61no—
92C2×C4×C8—61no—
93C2×C4×C8C2×D311yescharacteristic
94#384—31no—
95#340—61no—
96#331—62no—
97C23×D3—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—41no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—81no—
119C2×D12—81no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6C22⋊C412yes—
124C2×Dic6D812yes—
125C2×Dic6SD1612yes—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—22no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8SD1612yescomplemented
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—81no—
148C4⋊Dic3C22⋊C412yes—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156#204—48no—
157C22×Q8—31no—
158C22×D4—61no—
159C22×D4—61no—
160C22×D4—61no—
161C2×SD16—64no—
162C2×SD16—122no—
163C2×SD16—122no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—61no—
167C22×C8D1211yescharacteristic
168C22×C8D1211yescharacteristic
169#127—32no—
170#127—61no—
171#127—61no—
172#127—61no—
173C4×Q8—32no—
174C4×Q8—32no—
175C4×Q8—61no—
176C4×Q8—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—64no—
189C4⋊C8—61no—
190C4⋊C8C4×D312yes—
191#103—122no—
192#102—64no—
193#102—122no—
194#102—122no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—64no—
198C23×C3#12111yescharacteristic
199C22×D3—42no—
200C22×D3—81no—
201C22×D3—81no—
202C22×D3—81no—
203C22×D3—81no—
204C22×D3—81no—
205C22×D3—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4C2×D811yescharacteristic
215C2×C3×C4C2×SD1611yescharacteristic
216C2×C3×C4C2×SD1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231D12—42no—
232D12—42no—
233D12—42no—
234D12—42no—
235D12—82no—
236D12—82no—
237D12—81no—
238D12—81no—
239D12—81no—
240D12—81no—
241Dic6—22no—
242Dic6—22no—
243Dic6—22no—
244Dic6#10214yescomplemented
245C3×C8—22no—
246C3×C8—42no—
247C3×C8—41no—
248C3×C8—81no—
249C3⋊C8—42no—
250C24—121no—
251C2×Q8—32no—
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—241no—
272C2×D4—241no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4C2×D1211yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23911yescharacteristic
279SD16—128no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8#20412yescomplemented
295C4⋊C4—32no—
296C4⋊C4—61no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—241no—
304C42—62no—
305C42—62no—
306C42C2×C4×D311yescharacteristic
307C42#23511yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C22×C3#38611yescharacteristic
311C22×C3#46011yescharacteristic
312C22×C3#46411yescharacteristic
313C22×C3#46511yescharacteristic
314C22×C3#46611yescharacteristic
315C22×C3#47111yescharacteristic
316C22×C3#47211yescharacteristic
317C2×D3—82no—
318C2×D3—82no—
319C2×D3—82no—
320C2×D3—161no—
321C2×D3—161no—
322C2×D3—161no—
323C2×D3—161no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4C2×#10211yescharacteristic
327C3×C4#41611yescharacteristic
328C3×C4C4×SD1611yescharacteristic
329C3×C4#44011yescharacteristic
330C3×C4#44811yescharacteristic
331C3×C4#44911yescharacteristic
332C3×C4#46511yescharacteristic
333C3×C4#46611yescharacteristic
334Dic3—24no—
335Dic3—42no—
336Dic3—161no—
337C23—122no—
338C23—241no—
339C23—241no—
340C23—241no—
341C23—241no—
342C23—241no—
343C23—241no—
344C23#89511yescharacteristic
345Q8—34no—
346Q8—62no—
347Q8—62no—
348Q8—62no—
349D4—122no—
350D4—122no—
351D4—122no—
352D4—122no—
353D4—242no—
354D4—242no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4—241no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—241no—
368C2×C4—241no—
369C2×C4—241no—
370C2×C4C4×D1211yescharacteristic
371C2×C4D3×C22⋊C411yescharacteristic
372C2×C4#88211yescharacteristic
373C2×C4#88411yescharacteristic
374C2×C4#89311yescharacteristic
375C2×C4#89411yescharacteristic
376C2×C4C2×#20411yescharacteristic
377C2×C4#90411yescharacteristic
378C2×C4#90811yescharacteristic
379C2×C4#90911yescharacteristic
380C2×C4D3×D811yescharacteristic
381C2×C4D3×SD1611yescharacteristic
382C2×C4#91611yescharacteristic
383C2×C4#91911yescharacteristic
384C8—22no—
385C8—41no—
386C8—42no—
387C8—81no—
388C8—122no—
389C2×C3#166611yescharacteristic
390C2×C3#166811yescharacteristic
391C2×C3#167811yescharacteristic
392C2×C3#168211yescharacteristic
393C2×C3#188011yescharacteristic
394C2×C3#189311yescharacteristic
395C2×C3#190711yescharacteristic
396D3—162no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22—481no—
401C22—481no—
402C22—481no—
403C22—481no—
404C22#466511yescharacteristic
405C22#484511yescharacteristic
406C22#484811yescharacteristic
407C22#486411yescharacteristic
408C22#488311yescharacteristic
409C22#489711yescharacteristic
410C22#489811yescharacteristic
411C4—21no—
412C4—21no—
413C4—64no—
414C4—122no—
415C4—481no—
416C4C4×#20411yescharacteristic
417C4#471011yescharacteristic
418C4#473411yescharacteristic
419C4#473811yescharacteristic
420C4D3×#10211yescharacteristic
421C4#477511yescharacteristic
422C4#480411yescharacteristic
423C4#480711yescharacteristic
424C3#922211yescharacteristic, complemented, Sylow
425C2—482no—
426C2#6854611yescharacteristic
427C2#6854811yescharacteristic
428C2#6858611yescharacteristic
429C2#6859011yescharacteristic
430C2#6868611yescharacteristic
431C2#6888311yescharacteristic
432C2#6894811yescharacteristic
subgroupG/H
centerC23#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1277D3
hypocenterC3#9222
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic6C2semidirect
3. #70485C2semidirect
6. C3×#1277C2semidirect
7. #67702C2semidirect
24. #4848C4semidirect
35. #1277D3semidirect
137. C2×C3×C8SD16semidirect
244. Dic6#102semidirect
294. C2×C8#204semidirect
424. C3#9222semidirect

Maximal subgroups

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

subgroupindex
C2×#48482
C2×C8×Dic62
#704852
C2×#44852
#697722
C3×#12772
#677022
#92223