Finite Group Databasebeta

← #10723700

Subgroups — #10723700

Order 768.

1614 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5229C212yesmaximal
3#72134C212yesmaximal
4#71565C214yesmaximal
5#71564C214yesmaximal
6#71534C212yesmaximal
7#14450—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C23×C3×Q8C2211yescharacteristic
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8C418yescomplemented
12C22×C3×C4⋊C4C2214yes—
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×C4×Dic3C2212yes—
16#5229—28no—
17C2×C3⋊#95C2214yes—
18C2×C3⋊#95C2214yes—
19C2×C3⋊#95C2214yes—
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C22×C4⋊Q8D311yescharacteristic
26C2×#391—32no—
27#1446—32no—
28#1441—34no—
29#1436—34no—
30#1434—32no—
31C22×C3×Q8—24no—
32C22×C3×Q8—41no—
33C22×C3×Q8—42no—
34C22×C3×Q8—41no—
35C22×C3×Q8—42no—
36C22×C3×Q8—41no—
37C22×C3×Q8—41no—
38C22×C3×Q8C2×C414yes—
39C22×C3×Q8C2×C414yes—
40C22×C3×Q8D418yes—
41C23×C3×C4C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×Dic3C2314yes—
46C23×Dic3C2314yes—
47C3×C4⋊Q8—232no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4C2×C4116yes—
51C2×C3×C42—24no—
52C2×C3×C42C2×C414yes—
53C2×C4⋊Dic3—24no—
54C2×C4⋊Dic3—42no—
55C2×C4⋊Dic3—42no—
56C2×C4×Dic3—24no—
57C2×C4×Dic3—42no—
58C2×C4×Dic3—42no—
59C3⋊#95—28no—
60C3⋊#95—28no—
61C3⋊#95—28no—
62C3⋊#95—24no—
63C3⋊#95—42no—
64C3⋊#95—44no—
65C3⋊#95—42no—
66C3⋊#95Q818yescomplemented
67C23×Q8C2×D311yescharacteristic
68C23×Q8C2×D311yescharacteristic
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8Dic318yescomplemented
71C22×C4⋊C4—32no—
72C22×C4⋊C4C2×D314yes—
73C22×C42—32no—
74C22×C42C2×D311yescharacteristic
75#391—68no—
76C2×#95—32no—
77C2×#95—34no—
78C2×#95—32no—
79C2×#95—34no—
80C2×#95—34no—
81C2×#95—32no—
82C2×#95—34no—
83C2×#95—32no—
84C24×C3C2411yescharacteristic
85C2×C3×Q8—28no—
86C2×C3×Q8—44no—
87C2×C3×Q8—44no—
88C2×C3×Q8—44no—
89C2×C3×Q8—44no—
90C2×C3×Q8—48no—
91C2×C3×Q8—48no—
92C2×C3×Q8—48no—
93C2×C3×Q8—48no—
94C2×C3×Q8C22⋊C4116yes—
95C22×C3×C4—24no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4C22×C414yes—
109C22×C3×C4C22×C412yes—
110C22×C3×C4C22×C414yes—
111C22×C3×C4C22×C414yes—
112C22×C3×C4C2×D414yes—
113C22×C3×C4C2×D414yes—
114C22×C3×C4C2×D414yes—
115C22×C3×C4C2×Q814yes—
116C22×C3×C4#4114yes—
117C22×Dic3—24no—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—24no—
123C22×Dic3—24no—
124C22×Dic3—24no—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—24no—
130C22×Dic3C2×Q818yes—
131C3×C4⋊C4—232no—
132C3×C42—28no—
133C4⋊Dic3—48no—
134C4×Dic3—48no—
135C22×Q8—24no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—42no—
141C22×Q8—41no—
142C22×Q8C2×Dic314yes—
143C22×Q8C2×Dic314yes—
144C22×Q8#6718yes—
145C23×C4—34no—
146C23×C4—34no—
147C23×C4C22×D312yes—
148C23×C4C22×D312yes—
149C23×C4C22×D312yes—
150C23×C4C22×D311yescharacteristic
151C4⋊Q8—232no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4—122no—
156C2×C4⋊C4—122no—
157C2×C4⋊C4C2×Dic3116yes—
158C2×C42—24no—
159C2×C42—64no—
160C2×C42—122no—
161C2×C42—122no—
162C2×C42C2×Dic314yes—
163#95—38no—
164#95—68no—
165#95—64no—
166#95—68no—
167#95—68no—
168#95—122no—
169#95—124no—
170#95—122no—
171C23×C3C23×C411yescharacteristic
172C23×C3C22×D412yes—
173C23×C3C22×Q812yes—
174C23×C3C2×#4112yes—
175C23×C3H(5,𝔽2)12yes—
176C23×C3H(5,𝔽2)11yescharacteristic
177C23×C3H(5,𝔽2)11yescharacteristic
178C23×C32−1+414yes—
179C3×Q8—416no—
180C3×Q8—416no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—24no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—24no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—28no—
194C2×C3×C4—28no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4C2×C22⋊C414yes—
201C2×C3×C4C2×C22⋊C418yes—
202C2×C3×C4C4×Q818yes—
203C2×Dic3—28no—
204C2×Dic3—28no—
205C2×Dic3—28no—
206C2×Dic3—44no—
207C2×Dic3—44no—
208C2×Dic3—44no—
209C2×Dic3—44no—
210C2×Dic3—44no—
211C2×Dic3—44no—
212C2×Dic3—44no—
213C2×Dic3—44no—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—44no—
217C24C23×D311yescharacteristic
218C2×Q8—28no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8#217116yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—38no—
242C22×C4—64no—
243C22×C4—64no—
244C22×C4—64no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—64no—
249C22×C4—64no—
250C22×C4—64no—
251C22×C4—64no—
252C22×C4—64no—
253C22×C4—64no—
254C22×C4—64no—
255C22×C4D3×Q814yes—
256C22×C4#23914yes—
257C22×C4C22×Dic314yes—
258C22×C4C22×Dic314yes—
259C22×C4C22×Dic312yes—
260C22×C4C22×Dic314yes—
261C22×C4C2×#6714yes—
262C22×C4C2×#6714yes—
263C22×C4C2×#6714yes—
264C4⋊C4—232no—
265C4⋊C4—128no—
266C42—28no—
267C42—128no—
268C22×C3C22×C22⋊C411yescharacteristic
269C22×C3C2×C4×Q812yes—
270C22×C3#51811yescharacteristic
271C22×C3#51811yescharacteristic
272C22×C3#51912yes—
273C22×C3#53412yes—
274C22×C3#53412yes—
275C22×C3#54214yes—
276C22×C3#54414yes—
277C22×C3#54814yes—
278C22×C3D4×Q814yes—
279C22×C3#55412yes—
280C22×C3#55511yescharacteristic
281C22×C3Q8211yescharacteristic
282C22×C3#56312yes—
283C22×C3#56312yes—
284C3×C4—28no—
285C3×C4—24no—
286C3×C4—28no—
287C3×C4—44no—
288Dic3—48no—
289Dic3—84no—
290C23#100411yescharacteristic
291C23#100412yes—
292C23#100411yescharacteristic
293C23C2×D3×Q812yes—
294C23C2×#23912yes—
295C23#100714yes—
296C23C23×Dic311yescharacteristic
297C23C22×#6712yes—
298Q8—416no—
299Q8—416no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—68no—
320C2×C4—68no—
321C2×C4—68no—
322C2×C4—124no—
323C2×C4—124no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4—124no—
327C2×C4—124no—
328C2×C4—124no—
329C2×C4—124no—
330C2×C4—124no—
331C2×C4—124no—
332C2×C4—124no—
333C2×C4Q8×Dic318yes—
334C2×C4C2×#21714yes—
335C2×C4C2×#21718yes—
336C2×C3#230811yescharacteristic
337C2×C3#233112yes—
338C2×C3Q8×C22⋊C412yes—
339C2×C3#236911yescharacteristic
340C2×C3#238111yescharacteristic
341C2×C3#272914yes—
342C2×C3#273112yes—
343C2×C3#273412yes—
344C22#572211yescharacteristic
345C22#572914yes—
346C22#573012yes—
347C22#573314yes—
348C22#573411yescharacteristic
349C22#573512yes—
350C22#573512yes—
351C22#579811yescharacteristic
352C22#579811yescharacteristic
353C22#580412yes—
354C22#580412yes—
355C22C2×Q8×Dic312yes—
356C22#581212yes—
357C22Q8×#6714yes—
358C22#581714yes—
359C22C22×#21711yescharacteristic
360C4—24no—
361C4—28no—
362C4—28no—
363C4—44no—
364C4—128no—
365C4—244no—
366C3#1445011yescharacteristic, complemented, Sylow
367C2#8322812yes—
368C2#8323011yescharacteristic
369C2#8323111yescharacteristic
370C2#8324912yes—
371C2#8325012yes—
372C2#8326714yes—
373C2#8404311yescharacteristic
374C2Q8×#21712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×C4⋊Q8C2
socle, prime radical, cosocleC24×C3C24
hypercenterC22×C4⋊Q8D3
hypocenterC3#14450
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. C2×C3×C4⋊Q8C4semidirect
66. C3⋊#95Q8semidirect
70. C2×C4⋊Q8Dic3semidirect
366. C3#14450semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×#52292
#721342
#715652
#715642
#715342
#144503