Finite Group Databasebeta

← #10695929

Subgroups — #10695929

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×#3036C211yesmaximal, characteristic
2#83315C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#70500C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#463—22no—
14C3×#461—21no—
15C3×#459—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5221—21no—
19#5210—22no—
20#5094—21no—
21#5075—21no—
22#5040—21no—
23#5034—21no—
24C4×#214—22no—
25#4972—21no—
26#4949—21no—
27#4947—21no—
28#4936C2211yescharacteristic
29#4925C2211yescharacteristic
30#4925C2211yescharacteristic
31#4537—22no—
32D4⋊(C3⋊C8)—22no—
33#4484—22no—
34#3103—31no—
35#3036D311yescharacteristic
36#2580—31no—
37#1517—31no—
38#1487—31no—
39#1478—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×Dic6D411yescharacteristic
45C3×AGL(1,ℤ/8)—42no—
46C3×AGL(1,ℤ/8)—42no—
47C2×C3×SD16—22no—
48C2×C3×D8—22no—
49C2×C3×M16—21no—
50C3×C4⋊Q8D411yescharacteristic
51C3×#127D411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#217—22no—
65C2×#217—22no—
66C2×#214—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C4×Dic6—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yes—
78C3×#103—22no—
79C3×#102—22no—
80#832—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#810—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92C2×AGL(1,ℤ/8)—21no—
93#532C2×D311yescharacteristic
94C2×C4×Q8—31no—
95C2×C4×D4C2×D311yescharacteristic
96#481—61no—
97#475—61no—
98#463—22no—
99#461—21no—
100#460—61no—
101#459—21no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#423C2×D311yescharacteristic
107#417—21no—
108#399—61no—
109#393—61no—
110#390—61no—
111#389—61no—
112#385—62no—
113#382—31no—
114#327—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C2×C3×Q8—21no—
123C2×C3×D4—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—41no—
132C2×C3×D4SD1612yescomplemented
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—82no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C3×SD16—44no—
148C3×D8—44no—
149C3×M16—41no—
150C3×M16—41no—
151C2×C3×C8—22no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#217—44no—
165#217—82no—
166#214—44no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173Dic3⋊C4—82no—
174C4×Dic3—41no—
175C4×Dic3—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×#41—21no—
179C22×Q8—31no—
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—
188C4⋊Q8#6711yescharacteristic
189#127#6711yescharacteristic
190#124—21no—
191#121—21no—
192#121—21no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8#6712yes—
218#103—22no—
219#103—62no—
220#102—22no—
221#102—62no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C4×C8—32no—
225C4×C8—61no—
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3#12011yescharacteristic
236C3×Q8—41no—
237C3×Q8—41no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—42no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—81no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—82no—
254C2×C3×C4—82no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12711yescharacteristic
263C2×C3×C4C2×SD1611yescharacteristic
264C2×C3×C4#13511yescharacteristic
265C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
266C2×C3×C4#13711yescharacteristic
267C2×Dic3—22no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276Dic6—22no—
277Dic6—22no—
278Dic6—41no—
279Dic6—41no—
280C3×C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288C2×Q8—21no—
289C2×Q8—61no—
290C2×Q8—61no—
291C2×Q8—122no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4#21412yescomplemented
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—82no—
310C22×C4D3×D411yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4C2×#6711yescharacteristic
313SD16—44no—
314SD16—124no—
315D8—44no—
316M16—41no—
317M16—41no—
318C2×C8—22no—
319C2×C8—62no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—242no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—124no—
337C22⋊C4—242no—
338C42—121no—
339C42—121no—
340C42D3×D411yescharacteristic
341C42#23711yescharacteristic
342C42C2×#6711yescharacteristic
343C42C2×#6711yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—44no—
347C22×C3—42no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3#39211yescharacteristic
355C22×C3#44811yescharacteristic
356C22×C3#44911yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4#45011yescharacteristic
364C3×C4#45211yescharacteristic
365C3×C4#46511yescharacteristic
366C3×C4#46811yescharacteristic
367Dic3—42no—
368Dic3—82no—
369Dic3—81no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23#93711yescharacteristic
377Q8—41no—
378Q8—41no—
379Q8—62no—
380Q8—62no—
381Q8—121no—
382Q8—121no—
383D4—22no—
384D4—22no—
385D4—42no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—62no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—122no—
411C2×C4#93211yescharacteristic
412C2×C4C2×#21411yescharacteristic
413C2×C4#93711yescharacteristic
414C2×C4#93811yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#94011yescharacteristic
418C2×C4#95011yescharacteristic
419C2×C4#95111yescharacteristic
420C2×C4#95311yescharacteristic
421C8—42no—
422C8—122no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—82no—
427C2×C3—161no—
428C2×C3#200011yescharacteristic
429C2×C3#200811yescharacteristic
430C2×C3#201011yescharacteristic
431C22—21no—
432C22—21no—
433C22—44no—
434C22—42no—
435C22—82no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#503711yescharacteristic
442C22#504011yescharacteristic
443C22#522311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—122no—
451C4—242no—
452C4—241no—
453C4#504111yescharacteristic
454C4#504211yescharacteristic
455C4#521811yescharacteristic
456C4#523911yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—82no—
461C2—161no—
462C2#7168511yescharacteristic
463C2#7209911yescharacteristic
464C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#9998
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. #83315C2semidirect
3. #82388C2semidirect
4. #71814C2semidirect
5. #71597C2semidirect
6. #70500C2semidirect
7. #67713C2semidirect
132. C2×C3×D4SD16semidirect
301. C2×D4#214semidirect
457. C3#9998semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833152
#823882
#718142
#715972
#705002
#677132
#99983