Finite Group Databasebeta

← #10691339

Subgroups — #10691339

Order 768.

610 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4841C211yesmaximal, characteristic
2C2×C8×D12C211yesmaximal, characteristic
3#70479C211yesmaximal, characteristic
4C2×#4482C211yesmaximal, characteristic, complemented
5#69777C211yesmaximal, characteristic, complemented
6C3×#1277C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9217—31nomaximal, Sylow
9C22×#204—21no—
10C22×D3×C8—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#821—21no—
17C2×(C2×D3)⋊C8—21no—
18C2×#816—21no—
19#4935C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4841—22no—
22#4841C414yescomplemented
23C8×D12—22no—
24C8×D12—22no—
25#4551—21no—
26#4551—21no—
27#4551—21no—
28#4482—22no—
29C2×#461—31no—
30C2×C8×D4—31no—
31#1990—31no—
32#1802—31no—
33C2×#332—31no—
34#1296—31no—
35#1277D311yescharacteristic, complemented
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C22×C3⋊C8—21no—
52C2×#204—24no—
53C2×#204—42no—
54C2×#204—42no—
55C2×D3×C8—24no—
56C2×D3×C8—42no—
57C2×D3×C8—42no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12C2×C412yes—
61C4×D12D412yes—
62#869—21no—
63#869—21no—
64#869—21no—
65#869C2×C412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8C2×C412yes—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#821—42no—
73(C2×D3)⋊C8—24no—
74#816—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C22×SD16—61no—
78C23×C8—61no—
79C2×C4⋊Q8—31no—
80C2×C4×D4—31no—
81#461—34no—
82#461—62no—
83C8×D4—62no—
84C8×D4—62no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—61no—
89C2×#102—61no—
90C2×C22⋊C8—61no—
91C2×C4×C8C2×D311yescharacteristic
92#384—31no—
93#340—61no—
94#340—61no—
95#340—61no—
96#332—62no—
97C23×D3—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12C22⋊C412yes—
108C2×D12D812yes—
109C2×D12SD1612yes—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×Dic6—22no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—22no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8SD1612yescomplemented
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—81no—
150C4⋊Dic3C22⋊C412yes—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154#204—48no—
155D3×C8—48no—
156C22×Q8—61no—
157C22×D4—31no—
158C23×C4—61no—
159C2×SD16—64no—
160C2×SD16—122no—
161C2×SD16—122no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—64no—
165C22×C8—61no—
166C22×C8—122no—
167C22×C8—122no—
168C22×C8D1211yescharacteristic
169C22×C8D1211yescharacteristic
170C4⋊Q8—32no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—32no—
190C4⋊C8—61no—
191C4⋊C8C4×D312yes—
192#103—64no—
193#102—122no—
194C22⋊C8—64no—
195C4×C8—21no—
196C4×C8—21no—
197C23×C3#12111yescharacteristic
198C22×D3—22no—
199C22×D3—22no—
200C22×D3—22no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×D811yescharacteristic
214C2×C3×C4C2×SD1611yescharacteristic
215C2×C3×C4C2×SD1611yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230D12—22no—
231D12—22no—
232D12—22no—
233D12#10214yes—
234C4×D3—42no—
235C4×D3—42no—
236C4×D3—42no—
237C4×D3—42no—
238Dic6—42no—
239Dic6—42no—
240Dic6—81no—
241Dic6—81no—
242C3×C8—22no—
243C3×C8—42no—
244C3×C8—41no—
245C3×C8—81no—
246C3⋊C8—42no—
247C24—61no—
248C2×Q8—62no—
249C2×Q8—121no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×D4—32no—
255C2×D4—32no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C2×D1211yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4#23911yescharacteristic
272SD16—128no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—128no—
286C2×C8—121no—
287C2×C8—121no—
288C2×C8#20412yescomplemented
289C4⋊C4—32no—
290C4⋊C4—61no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—241no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42C2×C4×D311yescharacteristic
303C42#23511yescharacteristic
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C22×C3#38611yescharacteristic
307C22×C3#46111yescharacteristic
308C22×C3#46311yescharacteristic
309C22×C3#46511yescharacteristic
310C22×C3#46611yescharacteristic
311C22×C3#47111yescharacteristic
312C22×C3#47311yescharacteristic
313C2×D3—24no—
314C2×D3—42no—
315C2×D3—42no—
316C2×D3—42no—
317C2×D3—42no—
318C2×D3—42no—
319C2×D3—42no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4C2×#10211yescharacteristic
323C3×C4#41611yescharacteristic
324C3×C4C4×SD1611yescharacteristic
325C3×C4#44011yescharacteristic
326C3×C4#45011yescharacteristic
327C3×C4#45211yescharacteristic
328C3×C4#46511yescharacteristic
329C3×C4#46611yescharacteristic
330Dic3—42no—
331Dic3—82no—
332Dic3—161no—
333C23—62no—
334C23—62no—
335C23—62no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23#89511yescharacteristic
341Q8—122no—
342Q8—122no—
343Q8—241no—
344Q8—241no—
345D4—34no—
346D4—62no—
347D4—62no—
348D4—62no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—62no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4C4×D1211yescharacteristic
365C2×C4D3×C22⋊C411yescharacteristic
366C2×C4#88211yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4#89311yescharacteristic
369C2×C4#89411yescharacteristic
370C2×C4C2×#20411yescharacteristic
371C2×C4#90411yescharacteristic
372C2×C4#90811yescharacteristic
373C2×C4#90911yescharacteristic
374C2×C4D3×D811yescharacteristic
375C2×C4D3×SD1611yescharacteristic
376C2×C4#91611yescharacteristic
377C2×C4#91911yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—81no—
382C8—122no—
383C2×C3#167111yescharacteristic
384C2×C3#167311yescharacteristic
385C2×C3#167911yescharacteristic
386C2×C3#168111yescharacteristic
387C2×C3#187711yescharacteristic
388C2×C3#189311yescharacteristic
389C2×C3#190811yescharacteristic
390D3—44no—
391C22—64no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22—122no—
398C22#466511yescharacteristic
399C22#484111yescharacteristic
400C22#484411yescharacteristic
401C22#486411yescharacteristic
402C22#488311yescharacteristic
403C22#489611yescharacteristic
404C22#489811yescharacteristic
405C4—21no—
406C4—21no—
407C4—122no—
408C4—242no—
409C4—481no—
410C4C4×#20411yescharacteristic
411C4#471011yescharacteristic
412C4#473111yescharacteristic
413C4#473311yescharacteristic
414C4D3×#10211yescharacteristic
415C4#477511yescharacteristic
416C4#480411yescharacteristic
417C4#480711yescharacteristic
418C3#921711yescharacteristic, complemented, Sylow
419C2—124no—
420C2#6853711yescharacteristic
421C2#6853911yescharacteristic
422C2#6857811yescharacteristic
423C2#6858011yescharacteristic
424C2#6867211yescharacteristic
425C2#6888311yescharacteristic
426C2#6894911yescharacteristic
subgroupG/H
centerC23#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC23×C3#121
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1277D3
hypocenterC3#9217
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
4. C2×#4482C2semidirect
5. #69777C2semidirect
6. C3×#1277C2semidirect
7. #67702C2semidirect
22. #4841C4semidirect
35. #1277D3semidirect
133. C2×C3×C8SD16semidirect
288. C2×C8#204semidirect
418. C3#9217semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
C2×C8×D122
#704792
C2×#44822
#697772
C3×#12772
#677022
#92173