Finite Group Databasebeta

← #10697565

Subgroups — #10697565

Order 768.

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

normal subgroups only

This group has 590 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#83271C211yesmaximal, characteristic, complemented
3#82384C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#70508C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10241—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#463—22no—
14C3×#461—21no—
15C3×#459—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5221—21no—
19#5212—22no—
20#5080—21no—
21#5070—21no—
22#5034—22no—
23#5008—22no—
24#4977—21no—
25#4947—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30D4⋊(C3⋊C8)—22no—
31#4486—22no—
32#3227—31no—
33#3036D311yescharacteristic
34#2563—31no—
35#1549—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×Dic6—21no—
43C3×AGL(1,ℤ/8)—42no—
44C3×AGL(1,ℤ/8)—42no—
45C2×C3×SD16—22no—
46C2×C3×D8—22no—
47C2×C3×M16—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—24no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D412yes—
66C2×C4×Dic3—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—22no—
74C3×#102—22no—
75#832—24no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—21no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#463—22no—
94#461—21no—
95#459—21no—
96#431—31no—
97#431—31no—
98#423C2×D311yescharacteristic
99#417—21no—
100#393—62no—
101#390—61no—
102#382—61no—
103#380—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#327—62no—
107D4⋊C8—62no—
108C24×C3—21no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C2×C3×Q8—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C2×C3×D4SD1612yes—
124C22×C3×C4—21no—
125C22×C3×C4—44no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—22no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—82no—
137C3×SD16—44no—
138C3×D8—44no—
139C3×M16—41no—
140C3×M16—41no—
141C2×C3×C8—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—84no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—42no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×#41—21no—
165C22×Q8—61no—
166C22×D4#6711yescharacteristic
167C23×C4—21no—
168AGL(1,ℤ/8)—42no—
169AGL(1,ℤ/8)—42no—
170C2×SD16—22no—
171C2×D8—22no—
172C2×M16—21no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8#6711yescharacteristic
178#127#6711yescharacteristic
179#124—21no—
180#121—21no—
181#121—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—64no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#106—64no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201#103—22no—
202#102—22no—
203#102—64no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C4×C8—61no—
207C4×C8—61no—
208#95—32no—
209#95—64no—
210C23×C3—22no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12011yescharacteristic
217C3×Q8—41no—
218C3×Q8—41no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—41no—
222C3×D4—41no—
223C3×D4—41no—
224C3×D4—42no—
225C3×D4—41no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—82no—
235C2×C3×C4—82no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4C2×SD1611yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—44no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253Dic6—82no—
254Dic6—82no—
255C3×C8—42no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263C2×Q8—21no—
264C2×Q8—121no—
265C2×Q8—121no—
266C2×Q8—242no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4#21412yes—
277C22×C4—21no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4C2×#6711yescharacteristic
287SD16—44no—
288D8—44no—
289M16—41no—
290M16—41no—
291C2×C8—22no—
292C2×C8—62no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—62no—
299C4⋊C4—62no—
300C4⋊C4—122no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—244no—
308C42—121no—
309C42—121no—
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C42C2×#6711yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—44no—
317C22×C3—42no—
318C22×C3—82no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3#39211yescharacteristic
325C22×C3#44811yescharacteristic
326C22×C3#44911yescharacteristic
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—81no—
331C3×C4—81no—
332C3×C4#45011yescharacteristic
333C3×C4#45211yescharacteristic
334C3×C4#45412yes—
335C3×C4#46511yescharacteristic
336C3×C4#46811yescharacteristic
337Dic3—82no—
338Dic3—82no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23#93711yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—242no—
349Q8—242no—
350D4—22no—
351D4—22no—
352D4—42no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—122no—
370C2×C4—124no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4#93211yescharacteristic
375C2×C4C2×#21411yescharacteristic
376C2×C4#93711yescharacteristic
377C2×C4#93811yescharacteristic
378C2×C4#93911yescharacteristic
379C2×C4#93911yescharacteristic
380C2×C4#94011yescharacteristic
381C2×C4#95011yescharacteristic
382C2×C4#95111yescharacteristic
383C2×C4#95311yescharacteristic
384C8—42no—
385C8—122no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—82no—
390C2×C3—161no—
391C2×C3#200011yescharacteristic
392C2×C3#200311yescharacteristic
393C2×C3#200811yescharacteristic
394C22—21no—
395C22—21no—
396C22—44no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22#503711yescharacteristic
405C22#504011yescharacteristic
406C22#522311yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—242no—
413C4—242no—
414C4#504111yescharacteristic
415C4#504211yescharacteristic
416C4#520112yes—
417C4#521811yescharacteristic
418C4#523911yescharacteristic
419C3#1024111yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—82no—
423C2—161no—
424C2#7168511yescharacteristic
425C2#7209611yescharacteristic
426C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10241
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. #83271C2semidirect
3. #82384C2semidirect
4. #71811C2semidirect
5. #71597C2semidirect
6. #70508C2semidirect
7. #67715C2semidirect
419. C3#10241semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#832712
#823842
#718112
#715972
#705082
#677152
#102413