Finite Group Databasebeta

← #10691323

Subgroups — #10691323

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5084C211yesmaximal, characteristic
2C2×#4834C211yesmaximal, characteristic
3C2×C3×#333C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic
6#67703C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9215—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×D3×C8—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#821—21no—
15C2×(C2×D3)⋊C8—21no—
16#5084—22no—
17#5084C414yescomplemented
18C2×#809—21no—
19C2×#808—21no—
20#4935C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4834—22no—
24#4834—22no—
25C3×#333—22no—
26#4551—21no—
27#4529—21no—
28#4529—21no—
29C2×#476—31no—
30C2×C8×D4—31no—
31#1990—31no—
32#1802—31no—
33C2×#333D311yescharacteristic, complemented
34#1296—31no—
35#1277—31no—
36C22×C3×Q8—21no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C22×C3⋊C8—21no—
54C22×C3⋊C8D411yescharacteristic
55C22×C3⋊C8D411yescharacteristic
56C2×D3×C8—24no—
57C2×D3×C8—42no—
58C2×D3×C8—42no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12C2×C412yes—
62C4×D12D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67#821—42no—
68(C2×D3)⋊C8—24no—
69#809—24no—
70#808—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yes—
73C4×C3⋊C8—21no—
74C4×C3⋊C8—21no—
75C23×C8—61no—
76C2×C4⋊Q8C2×D311yescharacteristic
77C2×C4×D4—31no—
78#476—34no—
79#476—62no—
80C8×D4—62no—
81C8×D4—62no—
82C2×#106—61no—
83C2×C4⋊C8—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#102—61no—
87C2×#102—61no—
88C2×C22⋊C8—61no—
89C2×C4×C8—31no—
90#384—31no—
91#340—61no—
92#340—61no—
93#340—61no—
94#333—22no—
95C23×D3—21no—
96C2×C3×Q8—22no—
97C2×C3×Q8—41no—
98C2×C3×Q8—81no—
99C2×C3×Q8—81no—
100C22×C3×C4—41no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—41no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12C22⋊C412yes—
110C2×D12SD1612yes—
111C2×D12Q1612yes—
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134C4⋊Dic3—21no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—81no—
140C4⋊Dic3C22⋊C412yes—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8SD1612yes—
150D3×C8—48no—
151C22×Q8—21no—
152C22×D4—31no—
153C23×C4—61no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—31no—
157C22×C8—31no—
158C22×C8—64no—
159C22×C8—61no—
160C22×C8—122no—
161C22×C8—122no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C4×D312yes—
166C4×D4—32no—
167C4×D4—32no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#106—64no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—32no—
183C4⋊C8—61no—
184C4⋊C8#6712yes—
185#102—64no—
186#102—122no—
187C22⋊C8—64no—
188C4×C8—61no—
189C4×C8—61no—
190C23×C3#12111yescharacteristic
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—22no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C3×Q8—82no—
199C3×Q8—82no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4C2×SD1611yescharacteristic
213C2×C3×C4C2×Q1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225D12—22no—
226D12—22no—
227D12—22no—
228D12#10314yes—
229C4×D3—42no—
230C4×D3—42no—
231C4×D3—42no—
232C4×D3—42no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—22no—
236C3⋊C8—42no—
237C3⋊C8—41no—
238C24—61no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—81no—
242C2×Q8—81no—
243C2×D4—32no—
244C2×D4—32no—
245C2×D4—32no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C22×C4—41no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C2×C8—22no—
262C2×C8—32no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—62no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—128no—
275C2×C8—121no—
276C2×C8—121no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—32no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—241no—
288C22⋊C4—62no—
289C22⋊C4—62no—
290C42C2×C4×D311yescharacteristic
291C42D3×D411yescharacteristic
292C42#23711yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3#38611yescharacteristic
295C22×C3#46111yescharacteristic
296C22×C3#46311yescharacteristic
297C22×C3#46511yescharacteristic
298C22×C3#46711yescharacteristic
299C22×C3#47111yescharacteristic
300C22×C3#47211yescharacteristic
301C2×D3—24no—
302C2×D3—42no—
303C2×D3—42no—
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—42no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—82no—
311C3×C4C2×#10311yescharacteristic
312C3×C4#41611yescharacteristic
313C3×C4C4×SD1611yescharacteristic
314C3×C4#44011yescharacteristic
315C3×C4#45011yescharacteristic
316C3×C4#45211yescharacteristic
317C3×C4#46511yescharacteristic
318C3×C4#46711yescharacteristic
319Dic3—42no—
320Dic3—161no—
321C23—62no—
322C23—62no—
323C23—62no—
324C23—121no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23#88411yescharacteristic
329Q8—82no—
330Q8—82no—
331D4—34no—
332D4—62no—
333D4—62no—
334D4—62no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—62no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4D3×C22⋊C411yescharacteristic
351C2×C4#67⋊C411yescharacteristic
352C2×C4#88311yescharacteristic
353C2×C4D3×SD1611yescharacteristic
354C2×C4D3×SD1611yescharacteristic
355C2×C4#91411yescharacteristic
356C2×C4#91611yescharacteristic
357C2×C4D3×Q1611yescharacteristic
358C2×C4#91811yescharacteristic
359C2×C4#91911yescharacteristic
360C2×C4#91911yescharacteristic
361C2×C4C4×#6711yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93811yescharacteristic
364C8—42no—
365C8—62no—
366C8—81no—
367C8—121no—
368C8—122no—
369C2×C3#167111yescharacteristic
370C2×C3#167511yescharacteristic
371C2×C3#167711yescharacteristic
372C2×C3#168311yescharacteristic
373C2×C3#187711yescharacteristic
374C2×C3#189211yescharacteristic
375C2×C3#190711yescharacteristic
376D3—44no—
377C22—64no—
378C22—122no—
379C22—122no—
380C22—122no—
381C22—122no—
382C22—122no—
383C22—122no—
384C22#474811yescharacteristic
385C22#474911yescharacteristic
386C22#480711yescharacteristic
387C22#481311yescharacteristic
388C22#481811yescharacteristic
389C22#481911yescharacteristic
390C22#495611yescharacteristic
391C4—21no—
392C4—21no—
393C4—82no—
394C4—122no—
395C4—481no—
396C4#478811yescharacteristic
397C4D3×#10311yescharacteristic
398C4#480411yescharacteristic
399C4#481711yescharacteristic
400C4#516911yescharacteristic
401C4#517111yescharacteristic
402C4#518711yescharacteristic
403C4#518811yescharacteristic
404C3#921511yescharacteristic, complemented, Sylow
405C2—124no—
406C2#6849611yescharacteristic
407C2#6849711yescharacteristic
408C2#6849811yescharacteristic
409C2#6849911yescharacteristic
410C2#7008811yescharacteristic
411C2#7027011yescharacteristic
412C2#7027611yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#333C2
socleC23×C3#121
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenterC2×#333D3
hypocenterC3#9215
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. C2×C3×#333C2semidirect
4. #71787C2semidirect
6. #67703C2semidirect
7. #67702C2semidirect
17. #5084C4semidirect
33. C2×#333D3semidirect
404. C3#9215semidirect

Maximal subgroups

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

subgroupindex
C2×#50842
C2×#48342
C2×C3×#3332
#717872
#704792
#677032
#677022
#92153