Finite Group Databasebeta

← #10691376

Subgroups — #10691376

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×#4830C211yesmaximal, characteristic, complemented
3C2×#4485C211yesmaximal, characteristic
4#69588C211yesmaximal, characteristic, complemented
5#69580C211yesmaximal, characteristic
6C3×#1296C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#9222—31nomaximal, Sylow
9C22×#204—21no—
10C2×#874C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C2×#821—21no—
16C2×Dic3⋊C8—21no—
17C2×C8×Dic3—21no—
18C2×#810—21no—
19C2×#809—21no—
20C2×#809—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4848—22no—
24#4848C414yescomplemented
25#4830—22no—
26#4830—22no—
27#4529—21no—
28#4485—22no—
29C2×#460—31no—
30C2×C8×Q8—31no—
31#1957—31no—
32#1803—31no—
33C2×#331—31no—
34#1296D311yescharacteristic, complemented
35#1277—31no—
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×Dic6D411yescharacteristic
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C22×C3⋊C8—21no—
50C22×C3⋊C8D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
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⋊C8C2×C412yes—
66C3×C4⋊C8D412yes—
67#821—24no—
68Dic3⋊C8—24no—
69C8×Dic3—24no—
70#810—42no—
71#809—42no—
72#809—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
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—31no—
93C2×C4×C8—61no—
94#384C2×D311yescharacteristic
95#340—61no—
96#331—62no—
97C23×D3—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—21no—
103C22×Dic3—21no—
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—22no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—81no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C4⋊Dic3—21no—
142C4⋊Dic3C22⋊C412yes—
143Dic3⋊C4—22no—
144Dic3⋊C4—22no—
145C4×Dic3—22no—
146C4×Dic3—22no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8SD1612yescomplemented
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—31no—
167C22×C8—31no—
168C22×C8—61no—
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—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—64no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8C4×D312yes—
190C4⋊C8D1212yes—
191#103—122no—
192#102—64no—
193#102—122no—
194#102—122no—
195C4×C8—64no—
196C4×C8—61no—
197C4×C8—61no—
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×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4C2×D811yescharacteristic
218C2×C3×C4C2×SD1611yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
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—42no—
246C3×C8—42no—
247C3⋊C8—22no—
248C3⋊C8—41no—
249C3⋊C8—81no—
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—41no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4#23511yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279SD16—128no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—32no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—121no—
295C4⋊C4—32no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—61no—
303C4⋊C4—81no—
304C42—62no—
305C42—62no—
306C42C2×C4×D311yescharacteristic
307C42C2×D1211yescharacteristic
308C42D3×D411yescharacteristic
309C42#23911yescharacteristic
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×C4—161no—
327C3×C4C2×#10211yescharacteristic
328C3×C4#41611yescharacteristic
329C3×C4C4×SD1611yescharacteristic
330C3×C4#44011yescharacteristic
331C3×C4#44811yescharacteristic
332C3×C4#44911yescharacteristic
333C3×C4#46511yescharacteristic
334C3×C4#46611yescharacteristic
335Dic3—24no—
336Dic3—42no—
337C23—122no—
338C23—241no—
339C23—241no—
340C23—241no—
341C23—241no—
342C23—241no—
343C23—241no—
344C23#88411yescharacteristic
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—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4C4×D1211yescharacteristic
371C2×C4D3×C22⋊C411yescharacteristic
372C2×C4#88211yescharacteristic
373C2×C4#89311yescharacteristic
374C2×C4#89411yescharacteristic
375C2×C4#89511yescharacteristic
376C2×C4D3×D811yescharacteristic
377C2×C4#91111yescharacteristic
378C2×C4D3×SD1611yescharacteristic
379C2×C4D3×SD1611yescharacteristic
380C2×C4#91611yescharacteristic
381C2×C4#91611yescharacteristic
382C2×C4#91811yescharacteristic
383C2×C4#91911yescharacteristic
384C8—42no—
385C8—42no—
386C8—62no—
387C8—121no—
388C8—241no—
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#474911yescharacteristic
406C22#475411yescharacteristic
407C22#476211yescharacteristic
408C22#477511yescharacteristic
409C22#479811yescharacteristic
410C22#480711yescharacteristic
411C4—21no—
412C4—21no—
413C4—64no—
414C4—122no—
415C4—161no—
416C4D3×#10211yescharacteristic
417C4#478111yescharacteristic
418C4#480411yescharacteristic
419C4#480611yescharacteristic
420C4#485211yescharacteristic
421C4#486411yescharacteristic
422C4#488311yescharacteristic
423C4#488911yescharacteristic
424C3#922211yescharacteristic, complemented, Sylow
425C2—482no—
426C2#6840911yescharacteristic
427C2#6846411yescharacteristic
428C2#6847011yescharacteristic
429C2#6851011yescharacteristic
430C2#6888311yescharacteristic
431C2#6890911yescharacteristic
432C2#6900611yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#9222
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
4. #69588C2semidirect
6. C3×#1296C2semidirect
7. #67703C2semidirect
24. #4848C4semidirect
34. #1296D3semidirect
155. C2×C3⋊C8SD16semidirect
244. Dic6#102semidirect
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×#48302
C2×#44852
#695882
#695802
C3×#12962
#677032
#92223