Finite Group Databasebeta

← #10900960

Subgroups — #10900960

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×#870C211yesmaximal, characteristic
2#82075C211yesmaximal, characteristic
3C2×#4722C211yesmaximal, characteristic
4C2×#4721C211yesmaximal, characteristic
5C3×#1817C211yesmaximal, characteristic
6#69925C211yesmaximal, characteristic
7#69924C211yesmaximal, characteristic
8#68690C214yesmaximal, complemented
9#68634C214yesmaximal, complemented
10#29643—31nomaximal, Sylow
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×Dic3⋊C4—21no—
17C2×#870—22no—
18C2×#870—24no—
19C2×#870—24no—
20C2×#870C2214yes—
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23#5212C2211yescharacteristic
24#5211—21no—
25#5210—21no—
26C2×#818C2212yes—
27C2×#817C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#4933—22no—
31#4929C2212yes—
32#4928C2212yes—
33#4722—24no—
34#4721—24no—
35C3×#336C2214yes—
36#4551C2214yes—
37#4551C2214yes—
38C22×#125—31no—
39C2×#484—31no—
40C2×#483—31no—
41#2855—31no—
42#2059—34no—
43#2026—34no—
44#1824—31no—
45#1823—31no—
46#1817D311yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C22×Dic6C2311yescharacteristic
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C2×C3×C42—22no—
56C2×C3×C42—22no—
57C2×C3×C42D412yes—
58C2×C3×C42C2312yes—
59C2×#217—21no—
60C2×#217—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3D414yes—
66C2×C4⋊Dic3D414yes—
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71#924—22no—
72#924—22no—
73C2×Dic3⋊C4—28no—
74C2×Dic3⋊C4—28no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—21no—
79#870—216no—
80#870—28no—
81#870—28no—
82C3×C22⋊C8—22no—
83C3×C22⋊C8—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88#818—24no—
89#817—24no—
90#816—24no—
91#816—24no—
92C2×#125—34no—
93C2×#125—64no—
94C2×#125—64no—
95C2×#125—62no—
96C2×C22⋊Q8—31no—
97C22×C4⋊C4—31no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—61no—
100C22×C4⋊C4—61no—
101C22×C42C2×D311yescharacteristic
102#484—64no—
103#483—64no—
104C2×#107—32no—
105C2×#106—32no—
106C2×#103—32no—
107C2×#103—32no—
108C2×C22⋊C8C2×D311yescharacteristic
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#336C2×D314yes—
119C24×C3C2×D411yescharacteristic
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—24no—
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—44no—
140C22×Dic3—42no—
141C22×Dic3—42no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—24no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—24no—
148C2×C3×C8—24no—
149C3×C42—24no—
150C3×C42—22no—
151C3×C42—22no—
152#217—42no—
153#217—42no—
154C4⋊Dic3—24no—
155C4⋊Dic3—24no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—24no—
161C4⋊Dic3C2×D414yes—
162C4⋊Dic3C2×D414yes—
163Dic3⋊C4—216no—
164Dic3⋊C4—216no—
165Dic3⋊C4—44no—
166Dic3⋊C4—42no—
167C22×Q8—31no—
168C23×C4—61no—
169C23×C4—61no—
170C23×C4D1211yescharacteristic
171C23×C4D1211yescharacteristic
172C23×C4C22×D311yescharacteristic
173C22×C8C22×D312yes—
174C22×C8C22×D312yes—
175#125—68no—
176#125—68no—
177#125—616no—
178C22⋊Q8—62no—
179C22⋊Q8—62no—
180C2×C4⋊C4—34no—
181C2×C4⋊C4—34no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—68no—
187C2×C4⋊C4—68no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—64no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42D1212yes—
201C2×C42C22×D312yes—
202#107—64no—
203#106—64no—
204#103—64no—
205#103—64no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#95—61no—
209#95—62no—
210#95—62no—
211#95—62no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3#12012yes—
215C23×C3#13512yes—
216C23×C3#13712yes—
217C23×C3C22×D411yescharacteristic
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—44no—
228C2×C3×C4—44no—
229C2×C3×C4#12012yes—
230C2×C3×C4C22×D411yescharacteristic
231C2×C3×C4C22×D411yescharacteristic
232C2×C3×C42−1+414yes—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—48no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—44no—
239C2×Dic3—48no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247Dic6—42no—
248Dic6—42no—
249C3×C8—42no—
250C3×C8—42no—
251C24C2×D1211yescharacteristic
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—64no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—64no—
262C22×C4—64no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—124no—
269C22×C4—122no—
270C22×C4—122no—
271C22×C4C2×D1212yes—
272C22×C4C2×D1212yes—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D412yes—
275C22×C4D3×D412yes—
276C22×C4C23×D311yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C4⋊C4—34no—
282C4⋊C4—34no—
283C4⋊C4—616no—
284C4⋊C4—616no—
285C4⋊C4—64no—
286C4⋊C4—64no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—64no—
292C4⋊C4—124no—
293C4⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—122no—
296C42—24no—
297C42—22no—
298C42—22no—
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—42no—
311C3×C4—42no—
312C3×C4#53612yes—
313C3×C4#53612yes—
314Dic3—44no—
315Dic3—44no—
316Dic3—81no—
317Dic3—82no—
318Dic3—81no—
319C23—22no—
320C23—22no—
321C23#88212yes—
322C23#90412yes—
323C23#90912yes—
324C23C22×D1211yescharacteristic
325Q8—122no—
326Q8—122no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—64no—
339C2×C4—64no—
340C2×C4—128no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—124no—
344C2×C4—128no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4#88212yes—
353C2×C4C2×D3×D411yescharacteristic
354C2×C4C2×D3×D411yescharacteristic
355C2×C4#101014yes—
356C8—42no—
357C8—42no—
358C2×C3—24no—
359C2×C3#261411yescharacteristic
360C2×C3C2×#45211yescharacteristic
361C2×C3#300511yescharacteristic
362C2×C3#323212yes—
363C2×C3#323612yes—
364C22—22no—
365C22—22no—
366C22—24no—
367C22—24no—
368C22#473314yes—
369C22C2×#88211yescharacteristic
370C22#552612yes—
371C22C2×#90411yescharacteristic
372C22C2×#90911yescharacteristic
373C22#574812yes—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—124no—
378C4—124no—
379C4—241no—
380C4—241no—
381C4—242no—
382C4#564812yes—
383C4#564812yes—
384C3#2964311yescharacteristic, complemented, Sylow
385C2—24no—
386C2#7413511yescharacteristic
387C2C2×#473311yescharacteristic
388C2#7863811yescharacteristic
389C2#7927412yes—
390C2#7932012yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1817D3
hypocenterC3#29643
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #68690C2semidirect
9. #68634C2semidirect
384. C3#29643semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
#820752
C2×#47222
C2×#47212
C3×#18172
#699252
#699242
#686902
#686342
#296433