Finite Group Databasebeta

← #10690823

Subgroups — #10690823

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 368 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×#4546C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67634C212yesmaximal, complemented
6#9127—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8C412yes—
9C22×C3×C42C2211yescharacteristic
10C23×C3⋊C8—21no—
11C3×#391C412yes—
12C3×#389—21no—
13C3×#389—21no—
14C3×#384C2211yescharacteristic
15C3×#380—21no—
16C3×#380—21no—
17C2×#830—21no—
18C2×C4⋊(C3⋊C8)—21no—
19C2×C4⋊(C3⋊C8)—24no—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4⋊(C3⋊C8)C2212yes—
22#4546—24no—
23#4524—21no—
24C22×C4⋊C8—31no—
25#2857D311yescharacteristic
26#1797—31no—
27C2×#333—32no—
28#1296—32no—
29C22×C3×Q8C2×C412yes—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×C22⋊Q8—44no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42—22no—
44C2×C3×C42—22no—
45C2×C3×C42C2×C411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C22×C3⋊C8—24no—
50C22×C3⋊C8—22no—
51C22×C3⋊C8—22no—
52C22×C3⋊C8—42no—
53C22×C3⋊C8—42no—
54C3×#95—22no—
55C3×#95—22no—
56#830—42no—
57C4⋊(C3⋊C8)—24no—
58C4⋊(C3⋊C8)—22no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)—22no—
61C4⋊(C3⋊C8)D414yes—
62C23×C8—61no—
63C2×C4⋊Q8C2×D311yescharacteristic
64C2×C22⋊Q8Dic312yes—
65C22×C42C2×D311yescharacteristic
66C2×C4⋊C8—32no—
67C2×C4⋊C8—32no—
68C2×C4⋊C8—64no—
69C2×C4⋊C8—61no—
70C2×C22⋊C8—61no—
71#391Dic312yes—
72#389—21no—
73#389—21no—
74#384C2×D311yescharacteristic
75#380—21no—
76#380—21no—
77#336—61no—
78#333—64no—
79C24×C3C22⋊C411yescharacteristic
80C2×C3×Q8—24no—
81C2×C3×Q8—42no—
82C2×C3×Q8—42no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—41no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4SD1612yes—
98C22×C3×C4SD1612yes—
99C22×C3×C4Q1612yes—
100C22×C3×C4Q1612yes—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C3×C4⋊C4—42no—
105C3×C4⋊C4—44no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C22⋊C4—44no—
113C3×C42—22no—
114C3×C42—21no—
115C3×C42—22no—
116C3×C42—21no—
117C3×C42C2×D412yes—
118C3×C42#4112yes—
119C2×C3⋊C8—24no—
120C2×C3⋊C8—48no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—42no—
123C2×C3⋊C8—42no—
124C2×C3⋊C8—42no—
125C2×C3⋊C8—42no—
126C22×Q8C2×Dic312yes—
127C23×C4C2×Dic311yescharacteristic
128C23×C4#6711yescharacteristic
129C23×C4#6711yescharacteristic
130C22×C8—64no—
131C22×C8—62no—
132C22×C8—62no—
133C22×C8—122no—
134C22×C8—122no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C22⋊Q8—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×Dic312yes—
144C2×C22⋊C4—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C2×Dic311yescharacteristic
148C2×C42#6711yescharacteristic
149C2×C42#6711yescharacteristic
150C2×C42C22×D311yescharacteristic
151C4⋊C8—34no—
152C4⋊C8—64no—
153C4⋊C8—62no—
154C4⋊C8—62no—
155C4⋊C8—62no—
156C22⋊C8—122no—
157#95—22no—
158#95—22no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3#10012yes—
164C23×C3#10312yes—
165C23×C3#10312yes—
166C23×C3C2×C22⋊C411yescharacteristic
167C3×Q8—84no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—44no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4#10312yes—
191C2×C3×C4#10312yes—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×C3×C4C2×SD1611yescharacteristic
197C2×C3×C4C2×SD1611yescharacteristic
198C2×C3×C4C2×Q1611yescharacteristic
199C2×C3×C4C2×Q1611yescharacteristic
200C2×C3×C4#13512yes—
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C3⋊C8—44no—
204C3⋊C8—82no—
205C24#21711yescharacteristic
206C2×Q8—24no—
207C2×Q8—42no—
208C2×Q8—42no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4#21512yes—
221C22×C4#21512yes—
222C22×C4#21612yes—
223C22×C4#21612yes—
224C22×C4#21711yescharacteristic
225C22×C4#21711yescharacteristic
226C22×C4#21711yescharacteristic
227C22×C4C22×Dic311yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C2×C8—64no—
231C2×C8—128no—
232C2×C8—122no—
233C2×C8—122no—
234C2×C8—122no—
235C2×C8—122no—
236C2×C8—122no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—44no—
246C42—22no—
247C42—21no—
248C42—21no—
249C42—22no—
250C42D3×D412yes—
251C42#23712yes—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3#33314yescomplemented
259C22×C3#38611yescharacteristic
260C22×C3C2×#10011yescharacteristic
261C22×C3#41311yescharacteristic
262C22×C3C2×#10311yescharacteristic
263C22×C3C2×#10311yescharacteristic
264C22×C3#41611yescharacteristic
265C22×C3#41711yescharacteristic
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—41no—
270C3×C4—82no—
271C3×C4—82no—
272C3×C4#45011yescharacteristic
273C3×C4#45111yescharacteristic
274C3×C4#46512yes—
275C3×C4#46712yes—
276C3×C4#48311yescharacteristic
277C3×C4#48411yescharacteristic
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#83312yes—
283C23#83512yes—
284C23#83512yes—
285C23C2×#21711yescharacteristic
286Q8—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4#83512yes—
310C2×C4#83512yes—
311C2×C4D4×Dic312yes—
312C2×C4#93511yescharacteristic
313C2×C4#93711yescharacteristic
314C2×C4#93912yes—
315C2×C4C2×#21511yescharacteristic
316C2×C4C2×#21511yescharacteristic
317C2×C4C2×#21611yescharacteristic
318C2×C4C2×#21611yescharacteristic
319C2×C4#94911yescharacteristic
320C2×C4#95012yes—
321C2×C4#95111yescharacteristic
322C8—124no—
323C8—242no—
324C2×C3—24no—
325C2×C3C2×#33311yescharacteristic
326C2×C3#154211yescharacteristic
327C2×C3#155011yescharacteristic
328C2×C3#155111yescharacteristic
329C2×C3#187211yescharacteristic
330C2×C3#189211yescharacteristic
331C2×C3#190011yescharacteristic
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22#454614yescomplemented
339C22C2×#83311yescharacteristic
340C22#522011yescharacteristic
341C22C2×#83511yescharacteristic
342C22C2×#83511yescharacteristic
343C22#523211yescharacteristic
344C22#523311yescharacteristic
345C22#523711yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—82no—
351C4—82no—
352C4#504311yescharacteristic
353C4#504411yescharacteristic
354C4#504611yescharacteristic
355C4#504812yes—
356C4#505011yescharacteristic
357C4#505112yes—
358C3#912711yescharacteristic, complemented, Sylow
359C2—24no—
360C2#7168311yescharacteristic
361C2#7169211yescharacteristic
362C2#7169611yescharacteristic
363C2C2×#454611yescharacteristic
364C2#7180711yescharacteristic
365C2#7182811yescharacteristic
366C2#7183011yescharacteristic
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#9127
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×#4546C2semidirect
5. #67634C2semidirect
258. C22×C3#333semidirect
338. C22#4546semidirect
358. C3#9127semidirect

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×#45462
#715302
#676342
#91273