Finite Group Databasebeta

← #10690838

Subgroups — #10690838

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C2×#4547C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67634C212yesmaximal, complemented
6#9130—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C22×C3×C42C2211yescharacteristic
10C3×#391—21no—
11C3×#389—21no—
12C3×#389C412yes—
13C3×#384C2211yescharacteristic
14C3×#380—21no—
15C3×#380C412yes—
16C2×#830—21no—
17C2×#830—21no—
18#4925—24no—
19C2×C4⋊(C3⋊C8)C2212yes—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4×C3⋊C8—21no—
22#4547—24no—
23#4524—21no—
24C2×#431—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—22no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95—22no—
54#830—24no—
55#830—42no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)—22no—
59C4⋊(C3⋊C8)D414yes—
60C4×C3⋊C8—24no—
61C2×C4⋊Q8C2×D311yescharacteristic
62C2×C22⋊Q8—21no—
63C22×C42C2×D311yescharacteristic
64#431—64no—
65C2×C4⋊C8—32no—
66C2×C4⋊C8—32no—
67C2×C22⋊C8—61no—
68C2×C22⋊C8—61no—
69C2×C4×C8—61no—
70#391—21no—
71#389—21no—
72#389Dic312yes—
73#384C2×D311yescharacteristic
74#380—21no—
75#380Dic312yes—
76#336—61no—
77#332—64no—
78C24×C3C22⋊C411yescharacteristic
79C2×C3×Q8—42no—
80C2×C3×Q8—41no—
81C2×C3×Q8—41no—
82C2×C3×Q8—42no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
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×C4D812yes—
98C22×C3×C4SD1612yes—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C3×C4⋊C4—24no—
103C3×C4⋊C4—42no—
104C3×C4⋊C4—44no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—42no—
110C3×C22⋊C4—44no—
111C3×C42—22no—
112C3×C42—21no—
113C3×C42—22no—
114C3×C42—21no—
115C3×C42C2×D412yes—
116C3×C42#4112yes—
117C2×C3⋊C8—24no—
118C2×C3⋊C8—42no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—42no—
123C22×Q8—21no—
124C23×C4C2×Dic311yescharacteristic
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C22×C8—62no—
128C22×C8—62no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C2×Dic312yes—
138C2×C4⋊C4C2×Dic312yes—
139C2×C22⋊C4—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42C2×Dic311yescharacteristic
143C2×C42#6711yescharacteristic
144C2×C42#6711yescharacteristic
145C2×C42C22×D311yescharacteristic
146C4⋊C8—34no—
147C4⋊C8—62no—
148C4⋊C8—62no—
149C4⋊C8—62no—
150C22⋊C8—64no—
151C22⋊C8—122no—
152C4×C8—64no—
153#95—22no—
154#95—22no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3#10212yes—
162C23×C3C2×C22⋊C411yescharacteristic
163C3×Q8—82no—
164C3×Q8—82no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—24no—
172C2×C3×C4—24no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—44no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4#10112yes—
186C2×C3×C4#10212yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12112yes—
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4C2×D811yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4#13512yes—
194C2×C3×C4#13512yes—
195C2×C3×C4#13712yes—
196C3⋊C8—44no—
197C3⋊C8—82no—
198C24#21711yescharacteristic
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4#21312yes—
215C22×C4#21412yes—
216C22×C4#21711yescharacteristic
217C22×C4#21711yescharacteristic
218C22×C4#21711yescharacteristic
219C22×C4C22×Dic311yescharacteristic
220C22×C4C2×#6711yescharacteristic
221C22×C4C2×#6711yescharacteristic
222C2×C8—64no—
223C2×C8—122no—
224C2×C8—122no—
225C2×C8—122no—
226C2×C8—122no—
227C2×C8—122no—
228C4⋊C4—24no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C22⋊C4—44no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42D3×D412yes—
242C42#23712yes—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3#38611yescharacteristic
251C22×C3C2×#10111yescharacteristic
252C22×C3#41311yescharacteristic
253C22×C3C2×#10211yescharacteristic
254C22×C3#41611yescharacteristic
255C22×C3#41611yescharacteristic
256C22×C3#41911yescharacteristic
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4—82no—
262C3×C4—82no—
263C3×C4#45212yes—
264C3×C4#46512yes—
265C3×C4#46612yes—
266C3×C4#48512yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#83212yes—
274C23C2×#21711yescharacteristic
275Q8—82no—
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4#83212yes—
298C2×C4#83612yes—
299C2×C4C2×#21311yescharacteristic
300C2×C4C2×#21411yescharacteristic
301C2×C4D4×Dic312yes—
302C2×C4#93511yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#93912yes—
305C2×C4#94212yes—
306C2×C4#95012yes—
307C2×C4#95012yes—
308C8—124no—
309C8—242no—
310C2×C3—42no—
311C2×C3#154011yescharacteristic
312C2×C3#154411yescharacteristic
313C2×C3#154611yescharacteristic
314C2×C3#155411yescharacteristic
315C2×C3#187311yescharacteristic
316C2×C3#189311yescharacteristic
317C2×C3#190111yescharacteristic
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22C2×#83211yescharacteristic
326C22#522011yescharacteristic
327C22#522511yescharacteristic
328C22C2×#83611yescharacteristic
329C22#523311yescharacteristic
330C22#523311yescharacteristic
331C22#523711yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—41no—
336C4—82no—
337C4—82no—
338C4#503812yes—
339C4#504112yes—
340C4#504512yes—
341C4#504712yes—
342C3#913011yescharacteristic, complemented, Sylow
343C2—42no—
344C2#7168111yescharacteristic
345C2#7169311yescharacteristic
346C2#7169811yescharacteristic
347C2#7177311yescharacteristic
348C2#7180411yescharacteristic
349C2#7182911yescharacteristic
350C2#7183211yescharacteristic
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#9130
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
342. C3#9130semidirect

Maximal subgroups

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

subgroupindex
C3×#28572
C2×#49252
C2×#45472
#715302
#676342
#91303