Finite Group Databasebeta

← #10690830

Subgroups — #10690830

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
1#82076C211yesmaximal, characteristic
2C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic
3C3×#1797C211yesmaximal, characteristic
4C2×#4482C212yesmaximal, complemented
5#67702C212yesmaximal, complemented
6#9129—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924—21no—
9C23×C3⋊C8—21no—
10C2×#869C2211yescharacteristic
11C2×C3×C4⋊C8C2212yes—
12C2×C3×C22⋊C8—21no—
13#5212—21no—
14#5212C412yes—
15#4935C2211yescharacteristic
16#4929—21no—
17#4928—21no—
18#4928C412yes—
19C2×C4⋊(C3⋊C8)—21no—
20C2×C4⋊(C3⋊C8)—24no—
21C2×C4⋊(C3⋊C8)C2212yes—
22C3×#336—21no—
23#4482—24no—
24C22×C4⋊C8—31no—
25#2857—31no—
26#1797D311yescharacteristic
27C2×#332—32no—
28#1296—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6—21no—
33C22×C3×C8—22no—
34C2×C3×C42—22no—
35C2×C3×C42—22no—
36C2×C3×C42C2×C411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×#217—22no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C412yes—
45C2×C4⋊Dic3C2×C412yes—
46#924—42no—
47#924—42no—
48C2×Dic3⋊C4—22no—
49C22×C3⋊C8—24no—
50C22×C3⋊C8—22no—
51C22×C3⋊C8—42no—
52C22×C3⋊C8—42no—
53#869—22no—
54#869—22no—
55C3×C4⋊C8—22no—
56C3×C4⋊C8—22no—
57C3×C22⋊C8—42no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60C4⋊(C3⋊C8)—24no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)D414yes—
63C23×C8—61no—
64C2×C4⋊Q8—31no—
65C2×C22⋊Q8—61no—
66C22×C42C2×D311yescharacteristic
67C2×C4⋊C8—32no—
68C2×C4⋊C8—64no—
69C2×C4⋊C8—61no—
70C2×C4⋊C8C2×D312yes—
71C2×C22⋊C8—21no—
72#391—61no—
73#389—32no—
74#389—61no—
75#384—31no—
76#380—32no—
77#380—61no—
78#336—21no—
79#332—64no—
80C24×C3C22⋊C411yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—41no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4D812yes—
94C22×C3×C4SD1612yes—
95C22×C3×C4SD1612yes—
96C22×C3×C4Q1612yes—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—22no—
101C22×Dic3—22no—
102C2×Dic6—42no—
103C2×Dic6—41no—
104C2×Dic6—41no—
105C2×Dic6—42no—
106C2×C3×C8—42no—
107C2×C3×C8—42no—
108C2×C3×C8—42no—
109C3×C42—22no—
110C3×C42—21no—
111C3×C42—22no—
112C3×C42—21no—
113C3×C42C2×D412yes—
114C3×C42#4112yes—
115#217—44no—
116C4⋊Dic3—24no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123C4⋊Dic3—42no—
124Dic3⋊C4—44no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—48no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—42no—
129C22×Q8—61no—
130C23×C4C4×D311yescharacteristic
131C23×C4D1211yescharacteristic
132C23×C4#6711yescharacteristic
133C22×C8—22no—
134C22×C8—64no—
135C22×C8—62no—
136C22×C8—122no—
137C22×C8—122no—
138C4⋊Q8—62no—
139C4⋊Q8—62no—
140C22⋊Q8—122no—
141C22⋊Q8—122no—
142C2×C4⋊C4—32no—
143C2×C4⋊C4—32no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C22⋊C4—62no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42C4×D311yescharacteristic
152C2×C42D1211yescharacteristic
153C2×C42#6711yescharacteristic
154C2×C42C22×D311yescharacteristic
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—34no—
158C4⋊C8—64no—
159C4⋊C8—62no—
160C22⋊C8—42no—
161#95—62no—
162#95—62no—
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
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—44no—
187C2×C3×C4#10212yes—
188C2×C3×C4#10312yes—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12112yes—
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4C2×D811yescharacteristic
194C2×C3×C4C2×SD1611yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4C2×Q1611yescharacteristic
197C2×C3×C4#13512yes—
198C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
199C2×C3×C4#13711yescharacteristic
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206Dic6—82no—
207Dic6—82no—
208C3×C8—82no—
209C3⋊C8—44no—
210C24(C2×D3)⋊C411yescharacteristic
211C2×Q8—121no—
212C2×Q8—122no—
213C2×Q8—121no—
214C2×Q8—122no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4#20412yes—
227C22×C4Dic1212yes—
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4#21312yes—
232C22×C4#21512yes—
233C22×C4C2×C4×D311yescharacteristic
234C22×C4C2×D1211yescharacteristic
235C22×C4C2×#6711yescharacteristic
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—64no—
240C2×C8—128no—
241C2×C8—122no—
242C2×C8—122no—
243C4⋊C4—64no—
244C4⋊C4—122no—
245C4⋊C4—124no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C22⋊C4—124no—
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#45011yescharacteristic
278C3×C4#45111yescharacteristic
279C3×C4#46512yes—
280C3×C4#46612yes—
281C3×C4#48311yescharacteristic
282C3×C4#48411yescharacteristic
283Dic3—82no—
284Dic3—82no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#80912yes—
290C23#81612yes—
291C23#82412yes—
292C23C2×(C2×D3)⋊C411yescharacteristic
293Q8—242no—
294Q8—242no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4#80912yes—
318C2×C4#81612yes—
319C2×C4#67⋊C412yes—
320C2×C4#88211yescharacteristic
321C2×C4#88611yescharacteristic
322C2×C4C2×#20411yescharacteristic
323C2×C4C2×Dic1211yescharacteristic
324C2×C4#90811yescharacteristic
325C2×C4#90911yescharacteristic
326C2×C4C2×#21311yescharacteristic
327C2×C4#93912yes—
328C2×C4C2×#21511yescharacteristic
329C2×C4#95012yes—
330C8—82no—
331C8—124no—
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#448214yescomplemented
347C22#495811yescharacteristic
348C22C2×#80911yescharacteristic
349C22#500211yescharacteristic
350C22C2×#81611yescharacteristic
351C22#513211yescharacteristic
352C22#513311yescharacteristic
353C22C2×#82411yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—242no—
359C4—242no—
360C4#472111yescharacteristic
361C4#472211yescharacteristic
362C4#473111yescharacteristic
363C4#473911yescharacteristic
364C4#503812yes—
365C4#504812yes—
366C3#912911yescharacteristic, complemented, Sylow
367C2—24no—
368C2#6951511yescharacteristic
369C2#6993011yescharacteristic
370C2#6998611yescharacteristic
371C2C2×#448211yescharacteristic
372C2#7050311yescharacteristic
373C2#7077611yescharacteristic
374C2#7082011yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1797D3
hypocenterC3#9129
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
4. C2×#4482C2semidirect
5. #67702C2semidirect
265. C22×C3#332semidirect
346. C22#4482semidirect
366. C3#9129semidirect

Maximal subgroups

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

subgroupindex
#820762
C22×C4⋊(C3⋊C8)2
C3×#17972
C2×#44822
#677022
#91293