Finite Group Databasebeta

← #10690833

Subgroups — #10690833

Order 768.

644 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
1C3×#2857C211yesmaximal, characteristic
2C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic
3C2×#4547C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67634C212yesmaximal, complemented
6#9129—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C22×C3×C42C2211yescharacteristic
10C23×C3⋊C8—21no—
11C3×#391—21no—
12C3×#389—21no—
13C3×#389C412yes—
14C3×#384C2211yescharacteristic
15C3×#380—21no—
16C3×#380C412yes—
17C2×#830—21no—
18C2×C4⋊(C3⋊C8)—21no—
19C2×C4⋊(C3⋊C8)—24no—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4⋊(C3⋊C8)C2212yes—
22#4547—24no—
23#4524—21no—
24C22×C4⋊C8—31no—
25#2857D311yescharacteristic
26#1797—31no—
27C2×#332—32no—
28#1296—32no—
29C22×C3×Q8—21no—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×C22⋊Q8—42no—
36C3×C22⋊Q8—42no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C42—22no—
45C2×C3×C42—22no—
46C2×C3×C42C2×C411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C22×C3⋊C8—24no—
51C22×C3⋊C8—22no—
52C22×C3⋊C8—22no—
53C22×C3⋊C8—42no—
54C22×C3⋊C8—42no—
55C3×#95—22no—
56C3×#95—22no—
57#830—42no—
58C4⋊(C3⋊C8)—24no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)—22no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)D414yes—
63C23×C8—61no—
64C2×C4⋊Q8C2×D311yescharacteristic
65C2×C22⋊Q8—21no—
66C22×C42C2×D311yescharacteristic
67C2×C4⋊C8—32no—
68C2×C4⋊C8—32no—
69C2×C4⋊C8—64no—
70C2×C4⋊C8—61no—
71C2×C22⋊C8—61no—
72#391—21no—
73#389—21no—
74#389Dic312yes—
75#384C2×D311yescharacteristic
76#380—21no—
77#380Dic312yes—
78#336—61no—
79#332—64no—
80C24×C3C22⋊C411yescharacteristic
81C2×C3×Q8—42no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8—42no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—41no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4D812yes—
100C22×C3×C4SD1612yes—
101C22×C3×C4SD1612yes—
102C22×C3×C4Q1612yes—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C3×C4⋊C4—24no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—44no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—44no—
116C3×C42—22no—
117C3×C42—21no—
118C3×C42—22no—
119C3×C42—21no—
120C3×C42C2×D412yes—
121C3×C42#4112yes—
122C2×C3⋊C8—24no—
123C2×C3⋊C8—48no—
124C2×C3⋊C8—42no—
125C2×C3⋊C8—42no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—42no—
129C22×Q8—21no—
130C23×C4C2×Dic311yescharacteristic
131C23×C4#6711yescharacteristic
132C23×C4#6711yescharacteristic
133C22×C8—64no—
134C22×C8—62no—
135C22×C8—62no—
136C22×C8—122no—
137C22×C8—122no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C22⋊Q8—42no—
141C22⋊Q8—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4C2×Dic312yes—
147C2×C4⋊C4C2×Dic312yes—
148C2×C22⋊C4—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42C2×Dic311yescharacteristic
152C2×C42#6711yescharacteristic
153C2×C42#6711yescharacteristic
154C2×C42C22×D311yescharacteristic
155C4⋊C8—34no—
156C4⋊C8—64no—
157C4⋊C8—62no—
158C4⋊C8—62no—
159C4⋊C8—62no—
160C22⋊C8—122no—
161#95—22no—
162#95—22no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3#10112yes—
168C23×C3#10212yes—
169C23×C3#10312yes—
170C23×C3C2×C22⋊C411yescharacteristic
171C3×Q8—82no—
172C3×Q8—82no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—44no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4#10212yes—
196C2×C3×C4#10312yes—
197C2×C3×C4C4×D412yes—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12112yes—
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4C2×D811yescharacteristic
202C2×C3×C4C2×SD1611yescharacteristic
203C2×C3×C4C2×SD1611yescharacteristic
204C2×C3×C4C2×Q1611yescharacteristic
205C2×C3×C4#13512yes—
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4#13711yescharacteristic
208C3⋊C8—44no—
209C3⋊C8—82no—
210C24#21711yescharacteristic
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4#21312yes—
227C22×C4#21412yes—
228C22×C4#21512yes—
229C22×C4#21612yes—
230C22×C4#21711yescharacteristic
231C22×C4#21711yescharacteristic
232C22×C4#21711yescharacteristic
233C22×C4C22×Dic311yescharacteristic
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6711yescharacteristic
236C2×C8—64no—
237C2×C8—128no—
238C2×C8—122no—
239C2×C8—122no—
240C2×C8—122no—
241C2×C8—122no—
242C2×C8—122no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C22⋊C4—44no—
253C42—22no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42D3×D412yes—
258C42#23712yes—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3#33214yescomplemented
266C22×C3#38611yescharacteristic
267C22×C3C2×#10111yescharacteristic
268C22×C3#41311yescharacteristic
269C22×C3C2×#10211yescharacteristic
270C22×C3C2×#10311yescharacteristic
271C22×C3#41611yescharacteristic
272C22×C3#41711yescharacteristic
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—41no—
277C3×C4—82no—
278C3×C4—82no—
279C3×C4#45011yescharacteristic
280C3×C4#45111yescharacteristic
281C3×C4#46512yes—
282C3×C4#46612yes—
283C3×C4#48311yescharacteristic
284C3×C4#48411yescharacteristic
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#83212yes—
290C23#83512yes—
291C23#83612yes—
292C23C2×#21711yescharacteristic
293Q8—82no—
294Q8—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4#83212yes—
318C2×C4#83512yes—
319C2×C4C2×#21311yescharacteristic
320C2×C4C2×#21411yescharacteristic
321C2×C4D4×Dic312yes—
322C2×C4#93511yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93912yes—
325C2×C4C2×#21511yescharacteristic
326C2×C4C2×#21611yescharacteristic
327C2×C4#94911yescharacteristic
328C2×C4#95012yes—
329C2×C4#95111yescharacteristic
330C8—124no—
331C8—242no—
332C2×C3—24no—
333C2×C3C2×#33211yescharacteristic
334C2×C3#154211yescharacteristic
335C2×C3#154711yescharacteristic
336C2×C3#155211yescharacteristic
337C2×C3#187211yescharacteristic
338C2×C3#189311yescharacteristic
339C2×C3#190111yescharacteristic
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#454714yescomplemented
347C22C2×#83211yescharacteristic
348C22#522011yescharacteristic
349C22C2×#83511yescharacteristic
350C22C2×#83611yescharacteristic
351C22#523211yescharacteristic
352C22#523311yescharacteristic
353C22#523711yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—82no—
359C4—82no—
360C4#503812yes—
361C4#504112yes—
362C4#504311yescharacteristic
363C4#504411yescharacteristic
364C4#504611yescharacteristic
365C4#505011yescharacteristic
366C3#912911yescharacteristic, complemented, Sylow
367C2—24no—
368C2#7168111yescharacteristic
369C2#7169211yescharacteristic
370C2#7169811yescharacteristic
371C2C2×#454711yescharacteristic
372C2#7180611yescharacteristic
373C2#7182811yescharacteristic
374C2#7183111yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2857C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2857D3
hypocenterC3#9129
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#4547C2semidirect
5. #67634C2semidirect
265. C22×C3#332semidirect
346. C22#4547semidirect
366. C3#9129semidirect

Maximal subgroups

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

subgroupindex
C3×#28572
C22×C4⋊(C3⋊C8)2
C2×#45472
#715302
#676342
#91293