Finite Group Databasebeta

← #10690822

Subgroups — #10690822

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
1C22×C3×C4⋊C8C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic
3#71530C211yesmaximal, characteristic
4C2×#4481C212yesmaximal, complemented
5#67702C212yesmaximal, complemented
6#9127—31nomaximal, Sylow
7C23×C3×C8—21no—
8C22×C3×C42C2211yescharacteristic
9C2×#924C412yes—
10C2×#869C2211yescharacteristic
11C2×C3×C4⋊C8—21no—
12C2×C3×C4⋊C8—24no—
13C2×C3×C4⋊C8C2212yes—
14#5212—21no—
15#5212—21no—
16C2×#830—21no—
17#4935C2211yescharacteristic
18#4929C412yes—
19#4928—21no—
20#4928—21no—
21C2×C4⋊(C3⋊C8)C2212yes—
22#4524—21no—
23#4481—24no—
24C22×C4⋊C8D311yescharacteristic
25#2857—31no—
26#1797—31no—
27C2×#333—32no—
28#1296—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6C2×C412yes—
33C22×C3×C8—24no—
34C22×C3×C8—22no—
35C22×C3×C8—42no—
36C22×C3×C8—42no—
37C2×C3×C42—22no—
38C2×C3×C42—22no—
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×#217—22no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C412yes—
49#924—44no—
50C2×Dic3⋊C4—22no—
51C22×C3⋊C8—22no—
52#869—22no—
53#869—22no—
54C3×C4⋊C8—24no—
55C3×C4⋊C8—22no—
56C3×C4⋊C8D414yes—
57C3⋊#95—22no—
58C3⋊#95—22no—
59#830—42no—
60C4⋊(C3⋊C8)—22no—
61C4⋊(C3⋊C8)—22no—
62C23×C8—21no—
63C2×C4⋊Q8—31no—
64C2×C22⋊Q8—32no—
65C22×C42C2×D311yescharacteristic
66C2×C4⋊C8—21no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—32no—
69C2×C4⋊C8C2×D312yes—
70C2×C22⋊C8—61no—
71#391—32no—
72#389—61no—
73#389—61no—
74#384—31no—
75#380—61no—
76#380—61no—
77#336—61no—
78#333—64no—
79C24×C3C22⋊C411yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—41no—
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4SD1612yes—
93C22×C3×C4SD1612yes—
94C22×C3×C4Q1612yes—
95C22×C3×C4Q1612yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—22no—
100C22×Dic3—22no—
101C2×Dic6—24no—
102C2×Dic6—42no—
103C2×Dic6—42no—
104C2×C3×C8—24no—
105C2×C3×C8—48no—
106C2×C3×C8—42no—
107C2×C3×C8—42no—
108C3×C42—22no—
109C3×C42—21no—
110C3×C42—22no—
111C3×C42—21no—
112C3×C42C2×D412yes—
113C3×C42#4112yes—
114#217—44no—
115C4⋊Dic3—41no—
116C4⋊Dic3—42no—
117C4⋊Dic3—41no—
118C4⋊Dic3—42no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121C4⋊Dic3—42no—
122Dic3⋊C4—44no—
123C2×C3⋊C8—42no—
124C2×C3⋊C8—42no—
125C2×C3⋊C8—42no—
126C22×Q8—32no—
127C23×C4C4×D311yescharacteristic
128C23×C4D1211yescharacteristic
129C23×C4#6711yescharacteristic
130C22×C8—24no—
131C22×C8—22no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—62no—
135C4⋊Q8—62no—
136C4⋊Q8—62no—
137C22⋊Q8—124no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C22⋊C4—62no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C4×D311yescharacteristic
148C2×C42D1211yescharacteristic
149C2×C42#6711yescharacteristic
150C2×C42C22×D311yescharacteristic
151C4⋊C8—24no—
152C4⋊C8—22no—
153C4⋊C8—62no—
154C4⋊C8—62no—
155C4⋊C8#6714yes—
156C22⋊C8—122no—
157#95—62no—
158#95—62no—
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
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—24no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—44no—
183C2×C3×C4#10312yes—
184C2×C3×C4#10312yes—
185C2×C3×C4C4×D412yes—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12112yes—
188C2×C3×C4#12311yescharacteristic
189C2×C3×C4C2×SD1611yescharacteristic
190C2×C3×C4C2×SD1611yescharacteristic
191C2×C3×C4C2×Q1611yescharacteristic
192C2×C3×C4C2×Q1611yescharacteristic
193C2×C3×C4#13512yes—
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202Dic6—84no—
203C3×C8—44no—
204C3⋊C8—82no—
205C24(C2×D3)⋊C411yescharacteristic
206C2×Q8—64no—
207C2×Q8—122no—
208C2×Q8—122no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4#20412yes—
221C22×C4Dic1212yes—
222C22×C4(C2×D3)⋊C411yescharacteristic
223C22×C4(C2×D3)⋊C411yescharacteristic
224C22×C4(C2×D3)⋊C411yescharacteristic
225C22×C4#21412yes—
226C22×C4#21612yes—
227C22×C4C2×C4×D311yescharacteristic
228C22×C4C2×D1211yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C2×C8—24no—
231C2×C8—48no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—122no—
235C2×C8—122no—
236C2×C8—122no—
237C4⋊C4—122no—
238C4⋊C4—124no—
239C4⋊C4—122no—
240C4⋊C4—121no—
241C4⋊C4—122no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—124no—
246C42—22no—
247C42—21no—
248C42—22no—
249C42—21no—
250C42#23512yes—
251C42C2×#6712yes—
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#45011yescharacteristic
271C3×C4#45111yescharacteristic
272C3×C4#46512yes—
273C3×C4#46712yes—
274C3×C4#48311yescharacteristic
275C3×C4#48411yescharacteristic
276Dic3—82no—
277Dic3—82no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#81012yes—
283C23#81612yes—
284C23#82312yes—
285C23C2×(C2×D3)⋊C411yescharacteristic
286Q8—244no—
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—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4#81012yes—
310C2×C4#81612yes—
311C2×C4C2×#20411yescharacteristic
312C2×C4#90412yes—
313C2×C4C2×Dic1211yescharacteristic
314C2×C4C4×#6712yes—
315C2×C4#92911yescharacteristic
316C2×C4#93012yes—
317C2×C4#93211yescharacteristic
318C2×C4C2×#21411yescharacteristic
319C2×C4#94211yescharacteristic
320C2×C4C2×#21611yescharacteristic
321C2×C4#95311yescharacteristic
322C8—44no—
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#448114yescomplemented
339C22#496611yescharacteristic
340C22C2×#81011yescharacteristic
341C22C2×#81611yescharacteristic
342C22#511311yescharacteristic
343C22C2×#82311yescharacteristic
344C22#513211yescharacteristic
345C22#521311yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—242no—
351C4—242no—
352C4#497011yescharacteristic
353C4#497111yescharacteristic
354C4#511412yes—
355C4#511612yes—
356C4#521811yescharacteristic
357C4#522811yescharacteristic
358C3#912711yescharacteristic, complemented, Sylow
359C2—24no—
360C2#7043311yescharacteristic
361C2C2×#448111yescharacteristic
362C2#7043711yescharacteristic
363C2#7043811yescharacteristic
364C2#7155911yescharacteristic
365C2#7197211yescharacteristic
366C2#7200211yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC22×C3×C4⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC22×C4⋊C8D3
hypocenterC3#9127
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#4481C2semidirect
5. #67702C2semidirect
258. C22×C3#333semidirect
338. C22#4481semidirect
358. C3#9127semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C82
#820762
#715302
C2×#44812
#677022
#91273