Finite Group Databasebeta

← #10695635

Subgroups — #10695635

Order 768.

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

normal subgroups only

This group has 636 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×#2504C211yesmaximal, characteristic, complemented
2#82625C211yesmaximal, characteristic
3C2×#4968C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic, complemented
5#71554C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9950—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#397—22no—
13C3×#390—21no—
14C3×#386—22no—
15C3×#382—21no—
16C3×C4×C22⋊C4C2211yescharacteristic
17#5217—21no—
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5038—22no—
22#5036—21no—
23#5033—21no—
24#4968—22no—
25#4968—22no—
26#4967—21no—
27C2×#809—21no—
28C2×#808—21no—
29C2×#806—22no—
30#4537—22no—
31#4529—22no—
32#3186—31no—
33C2×#481—31no—
34#2504D311yescharacteristic, complemented
35#1964—31no—
36#1824—31no—
37#1519—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4D412yes—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#213—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C22×C3⋊C8—21no—
70C22×C3⋊C8D412yes—
71C3×#95—24no—
72C3×#95—21no—
73#832—22no—
74C3⋊#95—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830D412yes—
79#809—24no—
80#809—22no—
81#808—24no—
82#807—22no—
83#806—44no—
84C2×#121—31no—
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4C2×D311yescharacteristic
87#481—62no—
88#481—62no—
89#480—61no—
90#480—61no—
91#466—62no—
92#447—61no—
93#447—61no—
94C2×#106—61no—
95C2×#102—61no—
96C2×#99—62no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8—31no—
99#397—22no—
100#390—21no—
101#386—22no—
102#382—21no—
103#380—31no—
104C4×C22⋊C4C2×D311yescharacteristic
105#340—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—42no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—82no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×#67—41no—
131C2×#67—41no—
132C2×#67—42no—
133C2×#67—42no—
134C22×Dic3—21no—
135C22×Dic3—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—84no—
150C3×C42—42no—
151#217—42no—
152#213—82no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8—42no—
162C22×D4—21no—
163C22×D4—31no—
164C22×D4—61no—
165C22×D4#6711yescharacteristic
166C23×C4—21no—
167C23×C4C22×D311yescharacteristic
168C2×D8—62no—
169C22×C8—32no—
170C22×C8—61no—
171#121—21no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6712yes—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4#6712yes—
193C2×C42—21no—
194#107—62no—
195#106—64no—
196#102—64no—
197#102—62no—
198#102—62no—
199#99—124no—
200C22⋊C8—32no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—24no—
205#95—21no—
206#95—61no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C23×C3#12711yescharacteristic
216C23×C3#13512yes—
217C23×C3AGL(1,ℤ/8)12yes—
218C22×D3—41no—
219C22×D3—42no—
220C22×D3—81no—
221C22×D3—81no—
222C3×D4—42no—
223C3×D4—81no—
224C3×D4—82no—
225C3×D4—81no—
226C3×D4—81no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—82no—
241C2×C3×C4—82no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246#67—82no—
247#67—81no—
248#67—81no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254D12—42no—
255D12—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C24—21no—
259C24—61no—
260C24C2×#6711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—122no—
271C2×D4—121no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—121no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—61no—
282C22×C4—82no—
283C22×C4—121no—
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290D8—242no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—122no—
294C2×C8—122no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—84no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—122no—
312C42—42no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—84no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#39211yescharacteristic
326C22×C3#44912yes—
327C22×C3#46312yes—
328C22×C3#49511yescharacteristic
329C22×C3#49611yescharacteristic
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—162no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—82no—
338C3×C4—82no—
339C3×C4—81no—
340C3×C4#45812yes—
341Dic3—81no—
342Dic3—161no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—42no—
349C23—121no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23#93811yescharacteristic
354C23#94011yescharacteristic
355C23#94912yes—
356C23#95012yes—
357C23#95311yescharacteristic
358D4—42no—
359D4—81no—
360D4—82no—
361D4—81no—
362D4—81no—
363D4—122no—
364D4—122no—
365D4—242no—
366D4—241no—
367D4—241no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4#93711yescharacteristic
389C2×C4#93711yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#93911yescharacteristic
392C8—122no—
393C8—122no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—161no—
399C2×C3#200811yescharacteristic
400C2×C3#201211yescharacteristic
401C2×C3#202211yescharacteristic
402D3—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—84no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—482no—
419C22#505712yes—
420C22#505811yescharacteristic
421C22#505911yescharacteristic
422C22#522311yescharacteristic
423C22#523812yes—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—82no—
428C4—81no—
429C4—82no—
430C4—241no—
431C4—481no—
432C4#505312yes—
433C3#995011yescharacteristic, complemented, Sylow
434C2—22no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—161no—
439C2—481no—
440C2#7170611yescharacteristic
441C2#7172811yescharacteristic
442C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2504C2
socleC22×C3#392
prime radical#82625C2
hypercenter#2504D3
hypocenterC3#9950
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. C3×#2504C2semidirect
3. C2×#4968C2semidirect
4. #71675C2semidirect
5. #71554C2semidirect
6. #69456C2semidirect
7. #67617C2semidirect
34. #2504D3semidirect
433. C3#9950semidirect

Maximal subgroups

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

subgroupindex
C3×#25042
#826252
C2×#49682
#716752
#715542
#694562
#676172
#99503