Finite Group Databasebeta

← #10690843

Subgroups — #10690843

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 380 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#9131—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—41no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C2×C3×Q8—41no—
86C2×C3×Q8—41no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—21no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4D812yes—
102C22×C3×C4SD1612yes—
103C22×C3×C4SD1612yes—
104C22×C3×C4Q1612yes—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C3×C4⋊C4—24no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—44no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—42no—
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—81no—
172C3×Q8—81no—
173C3×Q8—81no—
174C3×Q8—81no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—44no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4#10212yes—
198C2×C3×C4#10312yes—
199C2×C3×C4C4×D412yes—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12112yes—
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4C2×D811yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4C2×SD1611yescharacteristic
206C2×C3×C4C2×Q1611yescharacteristic
207C2×C3×C4#13512yes—
208C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C3×C4#13711yescharacteristic
210C3⋊C8—44no—
211C3⋊C8—82no—
212C24#21711yescharacteristic
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4#21312yes—
231C22×C4#21412yes—
232C22×C4#21512yes—
233C22×C4#21612yes—
234C22×C4#21711yescharacteristic
235C22×C4#21711yescharacteristic
236C22×C4#21711yescharacteristic
237C22×C4C22×Dic311yescharacteristic
238C22×C4C2×#6711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C2×C8—64no—
241C2×C8—128no—
242C2×C8—122no—
243C2×C8—122no—
244C2×C8—122no—
245C2×C8—122no—
246C2×C8—122no—
247C4⋊C4—24no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—44no—
255C42—22no—
256C42—21no—
257C42—21no—
258C42—22no—
259C42D3×D412yes—
260C42#23712yes—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3#33214yescomplemented
268C22×C3#38611yescharacteristic
269C22×C3C2×#10111yescharacteristic
270C22×C3#41311yescharacteristic
271C22×C3C2×#10211yescharacteristic
272C22×C3C2×#10311yescharacteristic
273C22×C3#41611yescharacteristic
274C22×C3#41711yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—41no—
279C3×C4—82no—
280C3×C4—82no—
281C3×C4#44711yescharacteristic
282C3×C4#44811yescharacteristic
283C3×C4#46512yes—
284C3×C4#46712yes—
285C3×C4#48011yescharacteristic
286C3×C4#48111yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#83212yes—
292C23#83512yes—
293C23#83612yes—
294C23C2×#21711yescharacteristic
295Q8—81no—
296Q8—81no—
297Q8—81no—
298Q8—81no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4#83212yes—
322C2×C4#83512yes—
323C2×C4C2×#21311yescharacteristic
324C2×C4C2×#21411yescharacteristic
325C2×C4D4×Dic312yes—
326C2×C4#93511yescharacteristic
327C2×C4#93711yescharacteristic
328C2×C4#93912yes—
329C2×C4C2×#21511yescharacteristic
330C2×C4C2×#21611yescharacteristic
331C2×C4#94911yescharacteristic
332C2×C4#95012yes—
333C2×C4#95111yescharacteristic
334C8—124no—
335C8—242no—
336C2×C3—24no—
337C2×C3C2×#33211yescharacteristic
338C2×C3#154011yescharacteristic
339C2×C3#155011yescharacteristic
340C2×C3#155211yescharacteristic
341C2×C3#187411yescharacteristic
342C2×C3#189211yescharacteristic
343C2×C3#190111yescharacteristic
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22#454714yescomplemented
351C22C2×#83211yescharacteristic
352C22#522011yescharacteristic
353C22C2×#83511yescharacteristic
354C22C2×#83611yescharacteristic
355C22#523211yescharacteristic
356C22#523311yescharacteristic
357C22#523711yescharacteristic
358C4—22no—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—82no—
363C4—82no—
364C4#503311yescharacteristic
365C4#503411yescharacteristic
366C4#503611yescharacteristic
367C4#504011yescharacteristic
368C4#504812yes—
369C4#505112yes—
370C3#913111yescharacteristic, complemented, Sylow
371C2—24no—
372C2#7167711yescharacteristic
373C2#7169611yescharacteristic
374C2#7169811yescharacteristic
375C2C2×#454711yescharacteristic
376C2#7180411yescharacteristic
377C2#7183011yescharacteristic
378C2#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#9131
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
267. C22×C3#332semidirect
350. C22#4547semidirect
370. C3#9131semidirect

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
#91313