Finite Group Databasebeta

← #10695931

Subgroups — #10695931

Order 768.

602 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2580C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#399—21no—
14C3×#393—21no—
15C3×#390—21no—
16C3×#389—21no—
17C3×#385—22no—
18C3×#382C2211yescharacteristic
19#5227—21no—
20#5027—21no—
21C4×#215—22no—
22#5020—21no—
23#5019—21no—
24#5016—22no—
25#5015—21no—
26#4968—21no—
27#4966—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32D4⋊(C3⋊C8)—22no—
33#4451—22no—
34#3103—31no—
35#3036—31no—
36#2580D311yescharacteristic, complemented
37#1517—31no—
38#1487—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×#235—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63C2×#215—22no—
64C2×#214—22no—
65#932—42no—
66#932—42no—
67C2×#213—22no—
68#930—21no—
69#930—21no—
70C2×#208—21no—
71#875—21no—
72#874D411yescharacteristic
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic
76C3×#95—22no—
77C3×#95—21no—
78C3×#95D412yes—
79#835—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#809—22no—
84#809—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92C2×AGL(1,ℤ/8)—61no—
93#532—31no—
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×D4C2×D311yescharacteristic
96#481—61no—
97#475—61no—
98#463—62no—
99#461—61no—
100#460—61no—
101#459—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#423—31no—
107#417—61no—
108#399—21no—
109#393—21no—
110#390—21no—
111#389—21no—
112#385—22no—
113#382C2×D311yescharacteristic
114#327—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4SD1612yescomplemented
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—82no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C2×#67—41no—
138#235—41no—
139#235—41no—
140#235—41no—
141#235—41no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—21no—
145C2×C4×D3—41no—
146C2×Dic6—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—82no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—82no—
160C3×C42—41no—
161C3×C42—41no—
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166#215—44no—
167#214—44no—
168#213—44no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173#208—41no—
174#208—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×#41—61no—
179C22×Q8#6711yescharacteristic
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—61no—
188C4⋊Q8—31no—
189#127—31no—
190#124—61no—
191#121—61no—
192#121—61no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213#106—62no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218#103—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C4×C8—32no—
225C4×C8—61no—
226#95—22no—
227#95—21no—
228#95#6712yes—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3#12011yescharacteristic
235C22×D3—41no—
236C3×Q8—22no—
237C3×Q8—22no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—41no—
243C3×D4—41no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—82no—
258C2×C3×C4—82no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×C3×C4C2×SD1611yescharacteristic
266C2×C3×C4#13511yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4#13711yescharacteristic
269#67—81no—
270#67—81no—
271C2×Dic3—41no—
272D12—42no—
273D12—41no—
274D12—41no—
275D12—81no—
276C4×D3—81no—
277C4×D3—81no—
278Dic6—41no—
279Dic6—41no—
280C3⋊C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284#41—121no—
285#41—121no—
286#41—121no—
287#41—121no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8—61no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—121no—
301C2×D4#21512yescomplemented
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—121no—
310C22×C4C2×#6711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313SD16—124no—
314SD16—124no—
315D8—124no—
316M16—121no—
317M16—121no—
318C2×C8—62no—
319C2×C8—62no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—82no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—82no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C42—41no—
339C42—41no—
340C42D3×D411yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C42#23711yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—44no—
347C22×C3—42no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#39211yescharacteristic
352C22×C3#44811yescharacteristic
353C22×C3#44911yescharacteristic
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—42no—
362C3×C4—82no—
363C3×C4—81no—
364C3×C4—81no—
365C3×C4#45011yescharacteristic
366C3×C4#45211yescharacteristic
367C3×C4#46511yescharacteristic
368C3×C4#46811yescharacteristic
369Dic3—81no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—121no—
376C23#95311yescharacteristic
377Q8—22no—
378Q8—22no—
379Q8—41no—
380Q8—41no—
381Q8—121no—
382Q8—121no—
383D4—22no—
384D4—22no—
385D4—41no—
386D4—41no—
387D4—122no—
388D4—121no—
389D4—121no—
390D4—241no—
391D4—241no—
392D4—241no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—121no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4#93711yescharacteristic
412C2×C4#93711yescharacteristic
413C2×C4#93811yescharacteristic
414C2×C4#93911yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#94011yescharacteristic
417C2×C4C2×#21511yescharacteristic
418C2×C4#94211yescharacteristic
419C2×C4#94911yescharacteristic
420C2×C4#95011yescharacteristic
421C8—122no—
422C8—122no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—82no—
427C2×C3#200011yescharacteristic
428C2×C3#200811yescharacteristic
429C2×C3#201011yescharacteristic
430D3—161no—
431C22—21no—
432C22—21no—
433C22—44no—
434C22—42no—
435C22—82no—
436C22—81no—
437C22—81no—
438C22—241no—
439C22—241no—
440C22—241no—
441C22#522311yescharacteristic
442C22#522711yescharacteristic
443C22#523811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—82no—
450C4—81no—
451C4—81no—
452C4—241no—
453C4#504611yescharacteristic
454C4#504711yescharacteristic
455C4#504811yescharacteristic
456C4#504911yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—82no—
461C2—481no—
462C2#7169911yescharacteristic
463C2#7171511yescharacteristic
464C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2580C2
socleC22×C3#392
prime radical#82307C2
hypercenter#2580D3
hypocenterC3#9998
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C3×#2580C2semidirect
2. #82442C2semidirect
4. #71622C2semidirect
5. #71597C2semidirect
6. #69545C2semidirect
7. #67631C2semidirect
36. #2580D3semidirect
126. C2×C3×D4SD16semidirect
301. C2×D4#215semidirect
457. C3#9998semidirect

Maximal subgroups

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

subgroupindex
C3×#25802
#824422
#823072
#716222
#715972
#695452
#676312
#99983