Finite Group Databasebeta

← #10900957

Subgroups — #10900957

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic
3#82810C211yesmaximal, characteristic
4C2×#5044C211yesmaximal, characteristic
5#71553C211yesmaximal, characteristic
6#70566C214yesmaximal, complemented
7#70232C214yesmaximal, complemented
8#69925C211yesmaximal, characteristic
9#69910C211yesmaximal, characteristic
10#29643—31nomaximal, Sylow
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3—21no—
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×C4×Dic3C2211yescharacteristic
17C2×#890—24no—
18C2×#890—24no—
19C2×#890—22no—
20C2×#890C2214yes—
21C3×#483—24no—
22C2×C3×#106C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25#5229C2212yes—
26C2×#835C2212yes—
27#5212C2211yescharacteristic
28C2×#830C2211yescharacteristic
29#5044—24no—
30#4978—22no—
31#4972C2212yes—
32C2×#807C2212yes—
33#4946—21no—
34#4943—21no—
35#4551C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×#125—31no—
39C2×#484—31no—
40C2×#483D311yescharacteristic
41#2855—31no—
42#2059—34no—
43#2026—34no—
44#1824—31no—
45#1823—31no—
46#1817—31no—
47C22×C3×Q8C2311yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C22×C3×C8C2312yes—
54C3×C22⋊Q8—22no—
55C3×C22⋊Q8—22no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D414yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4—21no—
63C2×#217—21no—
64C2×C4⋊Dic3—28no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—24no—
68C2×C4⋊Dic3D414yes—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4—28no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3D412yes—
77C2×C4×Dic3C2312yes—
78C22×C3⋊C8C2312yes—
79#890—216no—
80#890—28no—
81#890—28no—
82C3×#106—24no—
83C3×#103—24no—
84C3×C22⋊C8—22no—
85#835—24no—
86C3⋊#95—21no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—22no—
91#807—24no—
92C2×#125—34no—
93C2×#125—62no—
94C2×#125—64no—
95C2×#125—64no—
96C2×C22⋊Q8C2×D311yescharacteristic
97C22×C4⋊C4—31no—
98C22×C4⋊C4—61no—
99C22×C4⋊C4—61no—
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C42—31no—
102#484—64no—
103#483—24no—
104C2×#107—32no—
105C2×#106C2×D312yes—
106C2×#103—32no—
107C2×#103C2×D312yes—
108C2×C22⋊C8—31no—
109C2×C22⋊C8C2×D311yescharacteristic
110#391—32no—
111#389—32no—
112#387—62no—
113#385—61no—
114#381—61no—
115#380—31no—
116#340—34no—
117#340—34no—
118#336—34no—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—24no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—24no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—44no—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×C3×C8—22no—
146C2×C3×C8—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4C2×D414yes—
153C3×C22⋊C4—42no—
154#217—42no—
155C4⋊Dic3—216no—
156C4⋊Dic3—24no—
157C4⋊Dic3—22no—
158C4⋊Dic3—24no—
159C4⋊Dic3C2×D414yes—
160Dic3⋊C4—216no—
161Dic3⋊C4—44no—
162C4×Dic3—24no—
163C4×Dic3—22no—
164C4×Dic3—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—24no—
167C22×Q8C22×D311yescharacteristic
168C23×C4—21no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4—61no—
172C23×C4C22×D311yescharacteristic
173C22×C8—32no—
174C22×C8C22×D312yes—
175#125—68no—
176#125—616no—
177#125—68no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—34no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4—64no—
188C2×C4⋊C4—68no—
189C2×C4⋊C4—68no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4#6714yes—
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—61no—
198C2×C42—32no—
199C2×C42—32no—
200C2×C42—62no—
201C2×C42—62no—
202#107—64no—
203#106—24no—
204#103—24no—
205#103—64no—
206C22⋊C8—22no—
207C22⋊C8—62no—
208#95—62no—
209#95—62no—
210#95—62no—
211#95—61no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3#12012yes—
215C23×C3#13512yes—
216C23×C3#13712yes—
217C23×C3C22×D411yescharacteristic
218C3×Q8—42no—
219C3×Q8—42no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—48no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4#12012yes—
232C2×C3×C4C22×D411yescharacteristic
233C2×C3×C4C22×D411yescharacteristic
234C2×C3×C42−1+414yes—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—24no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—44no—
243C2×Dic3—48no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C3×C8—42no—
250C3⋊C8—42no—
251C24D3×D411yescharacteristic
252C2×Q8—21no—
253C2×Q8—24no—
254C2×Q8—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—64no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—122no—
272C22×C4—124no—
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D412yes—
275C22×C4C2×#6712yes—
276C22×C4C23×D311yescharacteristic
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—62no—
280C2×C8—64no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—34no—
286C4⋊C4—42no—
287C4⋊C4—616no—
288C4⋊C4—616no—
289C4⋊C4—64no—
290C4⋊C4—62no—
291C4⋊C4—64no—
292C4⋊C4—124no—
293C4⋊C4C2×#6714yes—
294C22⋊C4—42no—
295C22⋊C4—122no—
296C42—64no—
297C42—62no—
298C42—62no—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—24no—
302C22×C3—22no—
303C22×C3#45214yes—
304C22×C3C2×#12011yescharacteristic
305C22×C3#53612yes—
306C22×C3C2×#13511yescharacteristic
307C22×C3C2×#13711yescharacteristic
308C22×C3#57512yes—
309C3×C4—22no—
310C3×C4—44no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4#53612yes—
314C3×C4#53612yes—
315Dic3—42no—
316Dic3—42no—
317Dic3—44no—
318Dic3—82no—
319C23—22no—
320C23—22no—
321C23#91512yes—
322C23#91612yes—
323C23#93712yes—
324C23C2×D3×D411yescharacteristic
325Q8—42no—
326Q8—42no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—64no—
345C2×C4—124no—
346C2×C4—128no—
347C2×C4—124no—
348C2×C4—124no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4#93712yes—
353C2×C4C2×D3×D411yescharacteristic
354C2×C4#101014yes—
355C2×C4C22×#6711yescharacteristic
356C8—42no—
357C8—122no—
358C2×C3—24no—
359C2×C3#261411yescharacteristic
360C2×C3C2×#45211yescharacteristic
361C2×C3#300511yescharacteristic
362C2×C3#323212yes—
363C2×C3#323612yes—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22#517314yes—
369C22#564812yes—
370C22C2×#91511yescharacteristic
371C22C2×#91611yescharacteristic
372C22#577712yes—
373C22C2×#93711yescharacteristic
374C4—22no—
375C4—44no—
376C4—81no—
377C4—81no—
378C4—122no—
379C4—124no—
380C4—122no—
381C4—242no—
382C4#564812yes—
383C4#582512yes—
384C3#2964311yescharacteristic, complemented, Sylow
385C2—24no—
386C2#8141912yes—
387C2#8142012yes—
388C2#8286411yescharacteristic
389C2C2×#517311yescharacteristic
390C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#483C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#483D3
hypocenterC3#29643
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
6. #70566C2semidirect
7. #70232C2semidirect
384. C3#29643semidirect

Maximal subgroups

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

subgroupindex
C22×#8902
C2×C3×#4832
#828102
C2×#50442
#715532
#705662
#702322
#699252
#699102
#296433