Finite Group Databasebeta

← #10900934

Subgroups — #10900934

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic
3#82699C211yesmaximal, characteristic
4C2×#5034C211yesmaximal, characteristic
5#71553C211yesmaximal, characteristic, complemented
6#70618C214yesmaximal, complemented
7#70164C214yesmaximal, complemented
8#69925C211yesmaximal, characteristic, complemented
9#69910C211yesmaximal, characteristic, complemented
10#29640—31nomaximal, Sylow
11C2×C3×#121C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#935—21no—
14C22×C4⋊Dic3—21no—
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4—21no—
17C22×C4×Dic3C2211yescharacteristic
18C2×#890—24no—
19C2×#890—24no—
20C2×#890—22no—
21C2×#890C2214yes—
22C3×#480—24no—
23C2×C3×#107C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26#5221C2212yes—
27#5220—21no—
28C2×#832C2212yes—
29#5212C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#5034—24no—
32#4972C2212yes—
33C2×#808C2212yes—
34#4944—22no—
35#4551C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×#125—31no—
39C2×#481—31no—
40C2×#480D311yescharacteristic
41#2854—31no—
42#2059—34no—
43#2025—34no—
44#1824—31no—
45#1823—31no—
46#1817—31no—
47C22×C3×D4—21no—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C23×Dic3—21no—
52C23×Dic3D411yescharacteristic
53C23×Dic3D411yescharacteristic
54C22×C3×C8C2312yes—
55C3×#121—22no—
56C3×#121—22no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4D414yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4—21no—
63C2×#217—21no—
64C2×#217—22no—
65C2×#217—22no—
66#935—44no—
67C2×C4⋊Dic3—28no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3D414yes—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×Dic3⋊C4—28no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D412yes—
80C2×C4×Dic3C2312yes—
81C22×C3⋊C8C2312yes—
82#890—216no—
83#890—28no—
84#890—28no—
85C3×#107—24no—
86C3×#102—24no—
87C3×C22⋊C8—22no—
88#832—24no—
89C3⋊#95—21no—
90C3⋊#95—22no—
91#830—22no—
92#808—24no—
93C2×#125—34no—
94C2×#125—62no—
95C2×#125—64no—
96C2×#125—64no—
97C2×#123—61no—
98C2×#121C2×D311yescharacteristic
99C22×C4⋊C4—31no—
100C22×C4⋊C4—61no—
101C22×C4⋊C4—61no—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C42—31no—
104#481—64no—
105#480—24no—
106C2×#107C2×D312yes—
107C2×#106—32no—
108C2×#102—32no—
109C2×#102C2×D312yes—
110C2×C22⋊C8—31no—
111C2×C22⋊C8C2×D311yescharacteristic
112#390—32no—
113#389—32no—
114#388—62no—
115#386—61no—
116#380—31no—
117#340—34no—
118#340—34no—
119#336—34no—
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—24no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—24no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—42no—
146C22×Dic3—44no—
147C22×Dic3C2×D412yes—
148C22×Dic3C2×D412yes—
149C2×C3×C8—22no—
150C2×C3×C8—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—42no—
157#217—44no—
158#217—44no—
159#217—42no—
160C4⋊Dic3—216no—
161C4⋊Dic3—24no—
162C4⋊Dic3—22no—
163C4⋊Dic3—24no—
164C4⋊Dic3C2×D414yes—
165Dic3⋊C4—216no—
166Dic3⋊C4—44no—
167C4×Dic3—24no—
168C4×Dic3—22no—
169C4×Dic3—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—24no—
172C22×D4—21no—
173C22×D4C22×D311yescharacteristic
174C23×C4—21no—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—61no—
178C23×C4C22×D311yescharacteristic
179C22×C8—32no—
180C22×C8C22×D312yes—
181#125—68no—
182#125—616no—
183#125—68no—
184#123—124no—
185#121—22no—
186#121—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—68no—
195C2×C4⋊C4—68no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4#6714yes—
200C2×C4⋊C4C22×D311yescharacteristic
201C2×C4⋊C4C22×D312yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206C2×C42—32no—
207C2×C42—32no—
208C2×C42—62no—
209C2×C42—62no—
210#107—24no—
211#106—64no—
212#102—24no—
213#102—64no—
214C22⋊C8—22no—
215C22⋊C8—62no—
216#95—62no—
217#95—61no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3#12012yes—
226C23×C3#13512yes—
227C23×C3AGL(1,ℤ/8)12yes—
228C23×C3C22×D411yescharacteristic
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—84no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—48no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4#12012yes—
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C22×D411yescharacteristic
245C2×C3×C42−1+414yes—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—24no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—44no—
254C2×Dic3—48no—
255C2×Dic3—44no—
256C2×Dic3—44no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C3×C8—42no—
261C3⋊C8—42no—
262C24—21no—
263C24D3×D411yescharacteristic
264C2×D4—21no—
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—32no—
275C22×C4—32no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—64no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—124no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D412yes—
289C22×C4C2×#6712yes—
290C22×C4C23×D311yescharacteristic
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—62no—
294C2×C8—64no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—34no—
300C4⋊C4—616no—
301C4⋊C4—616no—
302C4⋊C4—64no—
303C4⋊C4—62no—
304C4⋊C4—64no—
305C4⋊C4—124no—
306C4⋊C4C2×#6714yes—
307C22⋊C4—42no—
308C22⋊C4—124no—
309C22⋊C4—124no—
310C22⋊C4—122no—
311C42—64no—
312C42—62no—
313C42—62no—
314C22×C3—22no—
315C22×C3—24no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—81no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3#44914yes—
324C22×C3C2×#12011yescharacteristic
325C22×C3#53612yes—
326C22×C3C2×#13511yescharacteristic
327C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C3#57512yes—
329C3×C4—22no—
330C3×C4—44no—
331C3×C4—81no—
332C3×C4#53612yes—
333C3×C4#53612yes—
334Dic3—42no—
335Dic3—42no—
336Dic3—44no—
337Dic3—82no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23#91112yes—
346C23#91212yes—
347C23#93712yes—
348C23C2×D3×D411yescharacteristic
349D4—42no—
350D4—42no—
351D4—84no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—48no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—64no—
369C2×C4—124no—
370C2×C4—128no—
371C2×C4—124no—
372C2×C4—124no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4#93712yes—
377C2×C4C2×D3×D411yescharacteristic
378C2×C4#101014yes—
379C2×C4C22×#6711yescharacteristic
380C8—42no—
381C8—122no—
382C2×C3—24no—
383C2×C3—161no—
384C2×C3#261411yescharacteristic
385C2×C3C2×#44911yescharacteristic
386C2×C3#300511yescharacteristic
387C2×C3#323212yes—
388C2×C3#323412yes—
389C22—22no—
390C22—24no—
391C22—24no—
392C22—22no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22#515814yes—
399C22#564812yes—
400C22C2×#91111yescharacteristic
401C22C2×#91211yescharacteristic
402C22#577712yes—
403C22C2×#93711yescharacteristic
404C4—22no—
405C4—44no—
406C4—81no—
407C4—122no—
408C4—124no—
409C4—122no—
410C4—242no—
411C4#564812yes—
412C4#582512yes—
413C3#2964011yescharacteristic, complemented, Sylow
414C2—24no—
415C2—161no—
416C2#8127512yes—
417C2#8127612yes—
418C2#8286411yescharacteristic
419C2C2×#515811yescharacteristic
420C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#480C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#480D3
hypocenterC3#29640
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
5. #71553C2semidirect
6. #70618C2semidirect
7. #70164C2semidirect
8. #69925C2semidirect
9. #69910C2semidirect
413. C3#29640semidirect

Maximal subgroups

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

subgroupindex
C22×#8902
C2×C3×#4802
#826992
C2×#50342
#715532
#706182
#701642
#699252
#699102
#296403