Finite Group Databasebeta

← #10695637

Subgroups — #10695637

Order 768.

636 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#82665C211yesmaximal, characteristic, complemented
2#82625C211yesmaximal, characteristic
3C2×#4735C211yesmaximal, characteristic, complemented
4C3×#1519C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6#69502C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9950—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5221—21no—
15#5220—22no—
16#5217—21no—
17C4×#217C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#821—21no—
20C2×#817—21no—
21#5038—22no—
22#5036—21no—
23#5033—21no—
24#4977—21no—
25#4967—21no—
26C2×#806—22no—
27#4953—22no—
28#4735—22no—
29#4735—22no—
30C3×#327—22no—
31#4529—22no—
32#3186—31no—
33C2×#481—31no—
34#2504—31no—
35#1964—31no—
36#1824—31no—
37#1519D311yescharacteristic, complemented
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C22×D12D411yescharacteristic
45C22×C3×C8—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×#217—24no—
55C2×#217D412yes—
56C2×#213—22no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3D412yes—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4×Dic3—21no—
70C22×C3⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#95—21no—
74#832—22no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#821—24no—
80#817—24no—
81#809—22no—
82#807—22no—
83#806—44no—
84C2×#121—31no—
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4—31no—
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⋊C8C2×D311yescharacteristic
99#397—62no—
100#390—61no—
101#386—62no—
102#382—61no—
103#380C2×D311yescharacteristic
104C4×C22⋊C4—31no—
105#340—62no—
106#327—22no—
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—42no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—42no—
129C2×#67—42no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—82no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C2×C3×C8—22no—
140C2×C3×C8—42no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—42no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—42no—
147#217—22no—
148#217—22no—
149#217—84no—
150#213—82no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4×Dic3—42no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C22×D4—21no—
163C22×D4—31no—
164C22×D4—61no—
165C22×D4D1211yescharacteristic
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C2×D8—62no—
169C22×C8—21no—
170C22×C8—32no—
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—32no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4D1211yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—64no—
193C2×C42—61no—
194#107—62no—
195#106—64no—
196#102—62no—
197#102—64no—
198#102—62no—
199#99—124no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8#6712yes—
204#95—21no—
205#95—64no—
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—41no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242#67—82no—
243#67—81no—
244#67—81no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—82no—
253C2×Dic3—82no—
254D12—42no—
255D12—42no—
256C3×C8—42no—
257C3⋊C8—42no—
258C24—21no—
259C24—61no—
260C24D3×D411yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
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—42no—
277C22×C4—41no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—242no—
284C22×C4C2×D1211yescharacteristic
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4C2×#6711yescharacteristic
290D8—242no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—62no—
294C2×C8—122no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—61no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—244no—
312C42—122no—
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—81no—
338C3×C4—161no—
339C3×C4#45812yes—
340Dic3—82no—
341Dic3—82no—
342Dic3—81no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—42no—
349C23—121no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23#88211yescharacteristic
354C23#88411yescharacteristic
355C23#91112yes—
356C23#91612yes—
357C23#94011yescharacteristic
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—82no—
363D4—122no—
364D4—122no—
365D4—241no—
366D4—241no—
367D4—242no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—124no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—242no—
387C2×C4—242no—
388C2×C4#88211yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4#93011yescharacteristic
391C2×C4#93711yescharacteristic
392C8—42no—
393C8—122no—
394C2×C3—22no—
395C2×C3—21no—
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#478112yes—
420C22#478712yes—
421C22#496011yescharacteristic
422C22#515411yescharacteristic
423C22#516811yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—161no—
429C4—242no—
430C4—242no—
431C4—241no—
432C4#474212yes—
433C3#995011yescharacteristic, complemented, Sylow
434C2—21no—
435C2—22no—
436C2—22no—
437C2—21no—
438C2—161no—
439C2—481no—
440C2#7003111yescharacteristic
441C2#7021411yescharacteristic
442C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#392
prime radical#82625C2
hypercenter#1519D3
hypocenterC3#9950
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
1. #82665C2semidirect
3. C2×#4735C2semidirect
4. C3×#1519C2semidirect
5. #69528C2semidirect
6. #69502C2semidirect
7. #67678C2semidirect
37. #1519D3semidirect
433. C3#9950semidirect

Maximal subgroups

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

subgroupindex
#826652
#826252
C2×#47352
C3×#15192
#695282
#695022
#676782
#99503