Finite Group Databasebeta

← #10720197

Subgroups — #10720197

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×#926C211yesmaximal, characteristic
3C2×#5229C212yesmaximal, complemented
4C2×C4×#217C211yesmaximal, characteristic
5#71564C214yesmaximal, complemented
6#71563C212yesmaximal, complemented
7#71548C214yesmaximal
8#14138—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C24×C3×C4C2211yescharacteristic
11C2×C3×C22⋊Q8—24no—
12C2×C3×C22⋊Q8—24no—
13C2×C3×C22⋊Q8C418yescomplemented
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C2×#926—24no—
20C2×#926—28no—
21C22×C4⋊Dic3C2212yes—
22C22×C4×Dic3C2212yes—
23C22×C4×Dic3C2212yes—
24#5229—28no—
25C4×#217—24no—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2218yes—
28C2×C3⋊#95C2218yes—
29C22×C22⋊Q8D311yescharacteristic
30C22×#117—31no—
31C2×#391—32no—
32C2×C4×C22⋊C4—31no—
33#1440—34no—
34#1436—34no—
35C4⋊#95—32no—
36C25×C3C2311yescharacteristic
37C22×C3×Q8—24no—
38C22×C3×Q8—22no—
39C22×C3×Q8—22no—
40C22×C3×Q8—24no—
41C22×C3×Q8C2×C414yes—
42C23×C3×C4—24no—
43C23×C3×C4—24no—
44C23×C3×C4C2×C414yes—
45C23×C3×C4D414yes—
46C23×C3×C4D414yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×Dic3C2314yes—
52C23×Dic3C2314yes—
53C3×C22⋊Q8—232no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C4⋊C4C2×C418yes—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4C2×C418yes—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—28no—
66C2×#217—28no—
67#926—232no—
68C2×C4⋊Dic3—24no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3D418yes—
71C2×C4×Dic3—24no—
72C2×C4×Dic3—28no—
73C2×C4×Dic3—24no—
74C2×C4×Dic3—24no—
75C2×C4×Dic3—24no—
76C3⋊#95—24no—
77C3⋊#95—216no—
78C3⋊#95—216no—
79C23×Q8C2×D311yescharacteristic
80C24×C4C2×D311yescharacteristic
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8Dic318yescomplemented
84C2×#117—68no—
85C2×#117—64no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4C2×D311yescharacteristic
88C22×C4⋊C4C2×D312yes—
89C22×C22⋊C4—32no—
90C22×C22⋊C4—32no—
91C22×C22⋊C4C2×D312yes—
92C22×C42—32no—
93C22×C42—32no—
94#391—68no—
95C4×C22⋊C4—64no—
96C2×#95—38no—
97C2×#95—32no—
98C2×#95—38no—
99C24×C3—22no—
100C24×C3—24no—
101C24×C3—22no—
102C24×C3—24no—
103C24×C3C22×C412yes—
104C24×C3C2×D412yes—
105C24×C3C2×D412yes—
106C24×C3C2411yescharacteristic
107C2×C3×Q8—28no—
108C2×C3×Q8—216no—
109C2×C3×Q8—44no—
110C2×C3×Q8—48no—
111C2×C3×Q8—44no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—216no—
115C22×C3×C4—28no—
116C22×C3×C4—216no—
117C22×C3×C4—28no—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—28no—
124C22×C3×C4—24no—
125C22×C3×C4C22⋊C4116yes—
126C22×C3×C4C22×C414yes—
127C22×C3×C4C22×C414yes—
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C2×D414yes—
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4#4114yes—
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—24no—
140C22×Dic3—28no—
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—28no—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3C2×D418yes—
147C22×Dic3#4118yes—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—216no—
150C3×C22⋊C4—216no—
151#217—216no—
152#217—216no—
153C4⋊Dic3—216no—
154C4×Dic3—216no—
155C4×Dic3—48no—
156C25C22×D311yescharacteristic
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8C2×Dic314yes—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—34no—
165C23×C4—34no—
166C23×C4C2×Dic314yes—
167C23×C4#6714yes—
168C23×C4#6714yes—
169C23×C4C22×D312yes—
170C23×C4C22×D312yes—
171C23×C4C22×D311yescharacteristic
172C23×C4C22×D311yescharacteristic
173C22⋊Q8—232no—
174#117—632no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—38no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—64no—
182C2×C4⋊C4C2×Dic314yes—
183C2×C4⋊C4C2×Dic318yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—68no—
187C2×C22⋊C4—68no—
188C2×C22⋊C4—64no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4C2×Dic318yes—
191C2×C42—68no—
192C2×C42—64no—
193C2×C42—64no—
194C2×C42—64no—
195C2×C42—64no—
196#95—616no—
197#95—616no—
198#95—64no—
199C23×C3—24no—
200C23×C3—24no—
201C23×C3—24no—
202C23×C3—22no—
203C23×C3—24no—
204C23×C3—22no—
205C23×C3—24no—
206C23×C3—24no—
207C23×C3—24no—
208C23×C3—24no—
209C23×C3—44no—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—44no—
213C23×C3C2×C22⋊C414yes—
214C23×C3C23×C411yescharacteristic
215C23×C3C22×D412yes—
216C23×C3C22×D411yescharacteristic
217C23×C3C22×D411yescharacteristic
218C23×C3C2×#4112yes—
219C23×C3C2×#4114yes—
220C23×C32−1+414yes—
221C3×Q8—416no—
222C2×C3×C4—24no—
223C2×C3×C4—22no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—232no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—28no—
230C2×C3×C4—28no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—44no—
235C2×C3×C4—44no—
236C2×C3×C4C2×C22⋊C414yes—
237C2×C3×C4C2×C22⋊C414yes—
238C2×C3×C4C4×D418yes—
239C2×C3×C4#12014yes—
240C2×C3×C4#12118yes—
241C2×C3×C4#12314yes—
242C2×Dic3—28no—
243C2×Dic3—28no—
244C2×Dic3—216no—
245C2×Dic3—28no—
246C2×Dic3—28no—
247C2×Dic3—44no—
248C2×Dic3—44no—
249C2×Dic3—44no—
250C2×Dic3—44no—
251C2×Dic3—44no—
252C2×Dic3—48no—
253C2×Dic3—44no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24C22×Dic312yes—
259C24C2×#6712yes—
260C24C2×#6712yes—
261C24C23×D311yescharacteristic
262C2×Q8—216no—
263C2×Q8—28no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—216no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—216no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—38no—
281C22×C4—38no—
282C22×C4—64no—
283C22×C4—64no—
284C22×C4—64no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—68no—
288C22×C4—68no—
289C22×C4—64no—
290C22×C4—64no—
291C22×C4—64no—
292C22×C4#217116yes—
293C22×C4D3×D414yes—
294C22×C4#23714yes—
295C22×C4C22×Dic314yes—
296C22×C4C22×Dic314yes—
297C22×C4C22×Dic312yes—
298C22×C4C22×Dic312yes—
299C22×C4C2×#6712yes—
300C22×C4C2×#6712yes—
301C22×C4C2×#6712yes—
302C22×C4C2×#6712yes—
303C4⋊C4—28no—
304C4⋊C4—216no—
305C4⋊C4—616no—
306C22⋊C4—216no—
307C22⋊C4—616no—
308C22⋊C4—616no—
309C42—616no—
310C42—128no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—48no—
315C22×C3—48no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3C22×C22⋊C411yescharacteristic
319C22×C3C2×C4×D412yes—
320C22×C3C4×#4112yes—
321C22×C3#51912yes—
322C22×C3C2×#12011yescharacteristic
323C22×C3C2×#12112yes—
324C22×C3C2×#12311yescharacteristic
325C22×C3#52512yes—
326C22×C3#53212yes—
327C22×C3#53612yes—
328C22×C3#53612yes—
329C22×C3#54714yes—
330C22×C3#54814yes—
331C22×C3#55214yes—
332C22×C3#55614yes—
333C3×C4—24no—
334C3×C4—44no—
335C3×C4—44no—
336C3×C4#38618yes—
337Dic3—48no—
338Dic3—48no—
339C23—24no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23C2×#21714yes—
354C23C2×D3×D412yes—
355C23C2×#23712yes—
356C23#100714yes—
357C23D3×#4114yes—
358C23C23×Dic311yescharacteristic
359C23C22×#6711yescharacteristic
360C23C22×#6711yescharacteristic
361Q8—416no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—232no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—68no—
377C2×C4—68no—
378C2×C4—68no—
379C2×C4—68no—
380C2×C4—616no—
381C2×C4—128no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4D4×Dic318yes—
389C2×C4#93514yes—
390C2×C4#93714yes—
391C2×C4#93918yes—
392C2×C4C2×#21714yes—
393C2×C4C2×#21714yes—
394C2×C3—44no—
395C2×C3C2×#38611yescharacteristic
396C2×C3#229711yescharacteristic
397C2×C3#231011yescharacteristic
398C2×C3#236214yes—
399C2×C3#241514yes—
400C2×C3#243212yes—
401C2×C3#249912yes—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—48no—
406C22—44no—
407C22—44no—
408C22—48no—
409C22#563114yes—
410C22#563314yes—
411C22#563914yes—
412C22#564114yes—
413C22C2×D4×Dic312yes—
414C22C2×#93511yescharacteristic
415C22C2×#93711yescharacteristic
416C22C2×#93912yes—
417C22#581212yes—
418C22#581612yes—
419C22#581612yes—
420C22Dic3×#4112yes—
421C22#582812yes—
422C22#583212yes—
423C22C22×#21711yescharacteristic
424C4—24no—
425C4—44no—
426C4—44no—
427C4—128no—
428C4—128no—
429C4#522018yes—
430C3#1413811yescharacteristic, complemented, Sylow
431C2—44no—
432C2#8274814yes—
433C2#8275912yes—
434C2#8277412yes—
435C2#8281414yes—
436C2C2×#522011yescharacteristic
437C2#8406211yescharacteristic
438C2#8408611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×C22⋊Q8C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#14138
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#5229C2semidirect
5. #71564C2semidirect
6. #71563C2semidirect
13. C2×C3×C22⋊Q8C4semidirect
83. C2×C22⋊Q8Dic3semidirect
430. C3#14138semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9262
C2×#52292
C2×C4×#2172
#715642
#715632
#715482
#141383