Finite Group Databasebeta

← #10691429

Subgroups — #10691429

Order 768.

570 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×#5027C211yesmaximal, characteristic
2C2×#5024C211yesmaximal, characteristic, complemented
3C3×#1803C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic
5#69580C211yesmaximal, characteristic
6#67703C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#9230—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#874C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—21no—
14C3×#384C2211yescharacteristic
15#5027—22no—
16#5027C414yescomplemented
17#5024—22no—
18#5024—22no—
19C2×#809—21no—
20C2×#809—21no—
21C2×#809—21no—
22C2×C4⋊(C3⋊C8)—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×#803—21no—
26C2×C4×C3⋊C8C2211yescharacteristic
27C3×#340—21no—
28#4485—22no—
29C2×#460—31no—
30C2×#446—31no—
31#1957—31no—
32#1803D311yescharacteristic, complemented
33C2×#331—31no—
34#1296—31no—
35#1277—31no—
36C22×C3×Q8D411yescharacteristic
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×C3×C8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8C2×C412yes—
44C3×C4×Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#215—24no—
53C2×#215—42no—
54C2×#215—42no—
55C22×C3⋊C8—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8D411yescharacteristic
58C22×C3⋊C8D411yescharacteristic
59#874—21no—
60#874—21no—
61#874—21no—
62#874C2×C412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×#103—42no—
66#809—24no—
67#809—42no—
68#809—42no—
69C4⋊(C3⋊C8)—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yes—
73C4⋊(C3⋊C8)D412yes—
74#803—24no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C22×SD16—61no—
78C2×#127—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#460—34no—
81#460—62no—
82#446—62no—
83#446—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8—61no—
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—21no—
89C2×#102—61no—
90C2×#102—61no—
91C2×#102—61no—
92C2×#97—61no—
93C2×C4×C8—31no—
94#384C2×D311yescharacteristic
95#340—21no—
96#331—62no—
97C23×D3—41no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—81no—
125C2×D12—81no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—81no—
137C3×C4⋊C4C22⋊C412yes—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#215—48no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8SD1612yescomplemented
157C22×Q8#6711yescharacteristic
158C22×D4—61no—
159C22×D4—61no—
160C22×D4—61no—
161C2×SD16—64no—
162C2×SD16—122no—
163C2×SD16—122no—
164C22×C8—21no—
165C22×C8—31no—
166C22×C8—31no—
167C22×C8—61no—
168C22×C8—61no—
169#127—32no—
170#127—61no—
171#127—61no—
172#127—61no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8C4×D312yes—
176C4×Q8D1212yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—32no—
188C4⋊C8—64no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—42no—
192#102—64no—
193#102—122no—
194#102—122no—
195#97—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—
206C3×Q8—22no—
207C3×Q8—22no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12311yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×C3×C4#13711yescharacteristic
235D12—42no—
236D12—42no—
237D12—42no—
238D12—42no—
239D12—82no—
240D12—82no—
241D12—81no—
242D12—81no—
243D12—81no—
244D12—81no—
245C3×C8—81no—
246C3⋊C8—22no—
247C3⋊C8—42no—
248C3⋊C8—41no—
249C3⋊C8—42no—
250C24—121no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8(C2×D3)⋊C412yes—
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—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279SD16—128no—
280C2×C8—32no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—62no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—121no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—81no—
303C4⋊C4(C2×D3)⋊C412yes—
304C42—22no—
305C42—22no—
306C42C2×C4×D311yescharacteristic
307C42C2×D1211yescharacteristic
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C22×C3#38611yescharacteristic
311C22×C3#46011yescharacteristic
312C22×C3#46411yescharacteristic
313C22×C3#46811yescharacteristic
314C22×C3#46911yescharacteristic
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—42no—
327C3×C4—42no—
328C3×C4—161no—
329C3×C4#41711yescharacteristic
330C3×C4#41811yescharacteristic
331C3×C4C4×SD1611yescharacteristic
332C3×C4#44011yescharacteristic
333C3×C4#44811yescharacteristic
334C3×C4#44911yescharacteristic
335C3×C4#46811yescharacteristic
336C3×C4#46911yescharacteristic
337C23—122no—
338C23—241no—
339C23—241no—
340C23—241no—
341C23—241no—
342C23—241no—
343C23—241no—
344C23#93911yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
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—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4#67⋊C411yescharacteristic
371C2×C4#67⋊C411yescharacteristic
372C2×C4#88211yescharacteristic
373C2×C4#88611yescharacteristic
374C2×C4D3×SD1611yescharacteristic
375C2×C4#91411yescharacteristic
376C2×C4#91811yescharacteristic
377C2×C4#91911yescharacteristic
378C2×C4C2×(C2×D3)⋊C411yescharacteristic
379C2×C4#93211yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#94211yescharacteristic
382C2×C4#94911yescharacteristic
383C2×C4#94911yescharacteristic
384C8—62no—
385C8—81no—
386C8—122no—
387C8—121no—
388C8—122no—
389C2×C3#168611yescharacteristic
390C2×C3#168811yescharacteristic
391C2×C3#169811yescharacteristic
392C2×C3#170211yescharacteristic
393C2×C3#188011yescharacteristic
394C2×C3#189511yescharacteristic
395C2×C3#190711yescharacteristic
396D3—162no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22—481no—
401C22—481no—
402C22—481no—
403C22—481no—
404C22#495811yescharacteristic
405C22#503911yescharacteristic
406C22#504911yescharacteristic
407C22#516311yescharacteristic
408C22#518511yescharacteristic
409C22#519911yescharacteristic
410C22#520311yescharacteristic
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—161no—
416C4#478811yescharacteristic
417C4#480611yescharacteristic
418C4#481011yescharacteristic
419C4#481711yescharacteristic
420C4#496611yescharacteristic
421C4#500111yescharacteristic
422C4#503911yescharacteristic
423C4#504911yescharacteristic
424C3#923011yescharacteristic, complemented, Sylow
425C2—482no—
426C2#6951711yescharacteristic
427C2#7024011yescharacteristic
428C2#7030311yescharacteristic
429C2#7102511yescharacteristic
430C2#7105911yescharacteristic
431C2#7112011yescharacteristic
432C2#7115411yescharacteristic
subgroupG/H
centerC23#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1803C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1803D3
hypocenterC3#9230
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#5024C2semidirect
3. C3×#1803C2semidirect
6. #67703C2semidirect
7. #67634C2semidirect
16. #5027C4semidirect
32. #1803D3semidirect
156. C2×C3⋊C8SD16semidirect
424. C3#9230semidirect

Maximal subgroups

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

subgroupindex
C2×#50272
C2×#50242
C3×#18032
C2×#44852
#695802
#677032
#676342
#92303