Finite Group Databasebeta

← #10697669

Subgroups — #10697669

Order 768.

580 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3036C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6#70501C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10256—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#463—22no—
14C3×#461—21no—
15C3×#459—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5215—21no—
19C2×#830—22no—
20#5094—21no—
21#5075—21no—
22#5041—22no—
23#5019—21no—
24#5009—22no—
25#5008—21no—
26#5008—21no—
27#5007—21no—
28#4964—21no—
29#4924C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4484—22no—
34#3246—31no—
35#3077—31no—
36#3036D311yescharacteristic
37#1954—31no—
38#1493—31no—
39#1492—31no—
40#1292—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C3×AGL(1,ℤ/8)—42no—
45C3×AGL(1,ℤ/8)—42no—
46C2×C3×SD16—22no—
47C2×C3×D8—22no—
48C2×C3×M16—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×C3×C42C2311yescharacteristic
63C2×#214—22no—
64#926—21no—
65#924—21no—
66#924—21no—
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8D412yes—
70#870—21no—
71#869D411yescharacteristic
72#869D411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#103—22no—
78C3×#102—22no—
79#832—22no—
80#832—22no—
81#832—22no—
82#830—44no—
83#810—22no—
84#808—22no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yes—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×AGL(1,ℤ/8)—21no—
94#533—31no—
95#532C2×D311yescharacteristic
96C2×C4×D4C2×D311yescharacteristic
97#478—62no—
98#476—61no—
99#476—61no—
100#475—61no—
101#474—61no—
102#465—62no—
103#463—22no—
104#461—21no—
105#461—61no—
106#460—61no—
107#459—21no—
108#428—61no—
109#423—31no—
110#423C2×D311yescharacteristic
111C2×C4⋊C8—31no—
112#418—61no—
113#417—21no—
114C2×C22⋊C8—62no—
115C2×C4×C8—31no—
116Q8⋊C8—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C2×C3×Q8—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—41no—
133C2×C3×D4SD1612yescomplemented
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×Dic6—21no—
144C2×Dic6—21no—
145C3×SD16—44no—
146C3×D8—44no—
147C3×M16—41no—
148C3×M16—41no—
149C2×C3×C8—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162#217—41no—
163#214—82no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—42no—
179C2×#41—21no—
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—42no—
183AGL(1,ℤ/8)—42no—
184C2×SD16—22no—
185C2×SD16—62no—
186C2×D8—22no—
187C2×M16—21no—
188C2×M16—61no—
189C22×C8—32no—
190C22×C8—61no—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193C4⋊Q8#6711yescharacteristic
194#127#6711yescharacteristic
195#125—61no—
196#124—21no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199#121—21no—
200#121—21no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209#117—61no—
210C2×C4⋊C4#6711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214#106—62no—
215#106—62no—
216C4⋊C8—21no—
217C4⋊C8—32no—
218C4⋊C8—61no—
219C4⋊C8—61no—
220C4⋊C8—61no—
221C4⋊C8#6712yes—
222#103—22no—
223#103—62no—
224#102—22no—
225#102—62no—
226#102—62no—
227#102—62no—
228C22⋊C8—124no—
229C4×C8—61no—
230C4×C8—61no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3#12111yescharacteristic
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—42no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—81no—
250C3×D4—81no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—82no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12711yescharacteristic
268C2×C3×C4C2×SD1611yescharacteristic
269C2×C3×C4#13511yescharacteristic
270C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
271C2×C3×C4#13711yescharacteristic
272C2×Dic3—41no—
273C2×Dic3—41no—
274Dic6—42no—
275Dic6—41no—
276Dic6—41no—
277C3×C8—42no—
278C3⋊C8—42no—
279C3⋊C8—42no—
280C3⋊C8—42no—
281C24—21no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4#21412yescomplemented
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4D3×D411yescharacteristic
306C22×C4#23711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308SD16—44no—
309SD16—242no—
310D8—44no—
311M16—41no—
312M16—41no—
313M16—121no—
314M16—121no—
315C2×C8—22no—
316C2×C8—62no—
317C2×C8—62no—
318C2×C8—62no—
319C2×C8—121no—
320C2×C8—121no—
321C2×C8—122no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—121no—
337C42—121no—
338C42D3×D411yescharacteristic
339C42D3×D411yescharacteristic
340C42C2×#6711yescharacteristic
341C42C2×#6711yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3#39211yescharacteristic
353C22×C3#46511yescharacteristic
354C22×C3#46811yescharacteristic
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4#44811yescharacteristic
361C3×C4#44911yescharacteristic
362C3×C4#45011yescharacteristic
363C3×C4#45211yescharacteristic
364C3×C4#45311yescharacteristic
365C3×C4#45411yescharacteristic
366Dic3—81no—
367Dic3—81no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23#93911yescharacteristic
375Q8—41no—
376Q8—41no—
377Q8—122no—
378Q8—121no—
379Q8—121no—
380D4—22no—
381D4—22no—
382D4—42no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4#93211yescharacteristic
405C2×C4C2×#21411yescharacteristic
406C2×C4#93711yescharacteristic
407C2×C4#93711yescharacteristic
408C2×C4#93811yescharacteristic
409C2×C4#93911yescharacteristic
410C2×C4#94011yescharacteristic
411C2×C4#95011yescharacteristic
412C2×C4#95111yescharacteristic
413C2×C4#95311yescharacteristic
414C8—42no—
415C8—122no—
416C8—122no—
417C8—122no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—82no—
421C2×C3—161no—
422C2×C3#200011yescharacteristic
423C2×C3#200611yescharacteristic
424C2×C3#201711yescharacteristic
425C22—21no—
426C22—21no—
427C22—44no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#504111yescharacteristic
436C22#504211yescharacteristic
437C22#522311yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—241no—
444C4—241no—
445C4#503711yescharacteristic
446C4#504011yescharacteristic
447C4#519811yescharacteristic
448C4#520111yescharacteristic
449C4#521811yescharacteristic
450C4#523911yescharacteristic
451C3#1025611yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—82no—
455C2—161no—
456C2#7168511yescharacteristic
457C2#7209811yescharacteristic
458C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10256
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #83320C2semidirect
3. #82296C2semidirect
4. #71802C2semidirect
5. #71614C2semidirect
6. #70501C2semidirect
7. #67711C2semidirect
133. C2×C3×D4SD16semidirect
298. C2×D4#214semidirect
451. C3#10256semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833202
#822962
#718022
#716142
#705012
#677112
#102563