Finite Group Databasebeta

← #10690529

Subgroups — #10690529

Order 768.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C2×#4535C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67619C212yesmaximal, complemented
6#9073—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C3×#391C411yescharacteristic
11C3×#389C411yescharacteristic
12C3×#387—21no—
13C3×#385—21no—
14C3×#380C411yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C3×C4×C22⋊C4—21no—
18C2×#830—21no—
19C2×#830—21no—
20#4925—24no—
21C2×C4⋊(C3⋊C8)—21no—
22C2×C4⋊(C3⋊C8)C2212yes—
23C2×C4×C3⋊C8C2212yes—
24#4535—24no—
25#4524—21no—
26C2×#431—31no—
27C4×C22⋊Q8D311yescharacteristic
28#1797—31no—
29C2×Q8⋊C8—32no—
30#1279—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8C814yescomplemented
37C3×C4×Q8—22no—
38C3×C4×Q8—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—22no—
48C2×C3×C42—22no—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2×C411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42C2311yescharacteristic
55C22×C3⋊C8—22no—
56C22×C3⋊C8—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62#830—24no—
63#830—42no—
64C4⋊(C3⋊C8)—24no—
65C4⋊(C3⋊C8)—22no—
66C4⋊(C3⋊C8)—22no—
67C4×C3⋊C8—22no—
68C4×C3⋊C8D414yes—
69C2×C22⋊Q8Dic311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71C22×C42C2×D311yescharacteristic
72#431—64no—
73C2×C4⋊C8—32no—
74C2×C4⋊C8—61no—
75C2×C22⋊C8—61no—
76C2×C22⋊C8—61no—
77C2×C4×C8—32no—
78#391Dic311yescharacteristic
79#389Dic311yescharacteristic
80#387—21no—
81#385—21no—
82#380Dic311yescharacteristic
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2×D311yescharacteristic
85C4×C22⋊C4—21no—
86#336—61no—
87Q8⋊C8—64no—
88C24×C3C22⋊C411yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8—22no—
91C2×C3×Q8—22no—
92C2×C3×Q8C2×C812yes—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C2×C812yes—
111C22×C3×C4M1612yes—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C3×C4⋊C4—24no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—42no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4C2×C812yes—
123C3×C22⋊C4—24no—
124C3×C22⋊C4—42no—
125C3×C42—22no—
126C3×C42—21no—
127C3×C42—22no—
128C3×C42—21no—
129C3×C42—42no—
130C3×C42—41no—
131C3×C42—41no—
132C3×C42C2×D412yes—
133C3×C42#4112yes—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—42no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—42no—
140C22×Q8C2×Dic311yescharacteristic
141C23×C4C2×Dic311yescharacteristic
142C23×C4#6711yescharacteristic
143C23×C4#6711yescharacteristic
144C22×C8—62no—
145C22×C8—62no—
146C22⋊Q8—22no—
147C22⋊Q8C3⋊C814yescomplemented
148C4×Q8—22no—
149C4×Q8—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C2×Dic311yescharacteristic
154C2×C4⋊C4C2×Dic311yescharacteristic
155C2×C4⋊C4C2×Dic311yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42C2×Dic311yescharacteristic
163C2×C42#6711yescharacteristic
164C2×C42#6711yescharacteristic
165C2×C42C22×D311yescharacteristic
166C4⋊C8—64no—
167C4⋊C8—62no—
168C4⋊C8—62no—
169C22⋊C8—64no—
170C22⋊C8—122no—
171C4×C8—34no—
172C4×C8—62no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3C22⋊C812yes—
185C23×C3C2×C22⋊C411yescharacteristic
186C3×Q8—24no—
187C3×Q8—42no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4C22⋊C812yes—
217C2×C3×C4C4×D412yes—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12112yes—
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C22×C811yescharacteristic
222C2×C3×C4C2×M1611yescharacteristic
223C2×C3×C4#13112yes—
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4#13712yes—
226C2×C3×C4#13711yescharacteristic
227C3⋊C8—44no—
228C3⋊C8—82no—
229C24#21711yescharacteristic
230C2×Q8—21no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8C2×C3⋊C812yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4C2×C3⋊C812yes—
249C22×C4#20812yes—
250C22×C4#21711yescharacteristic
251C22×C4#21711yescharacteristic
252C22×C4#21711yescharacteristic
253C22×C4C22×Dic311yescharacteristic
254C22×C4C2×#6711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256C2×C8—64no—
257C2×C8—122no—
258C2×C8—122no—
259C2×C8—122no—
260C2×C8—122no—
261C2×C8—122no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4C2×C3⋊C812yes—
270C22⋊C4—24no—
271C22⋊C4—42no—
272C42—22no—
273C42—21no—
274C42—21no—
275C42—22no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42D3×D412yes—
280C42#23712yes—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3#38611yescharacteristic
289C22×C3C2×C22⋊C811yescharacteristic
290C22×C3#40811yescharacteristic
291C22×C3#41711yescharacteristic
292C22×C3#41911yescharacteristic
293C22×C3#42111yescharacteristic
294C22×C3#42111yescharacteristic
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—42no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4C8×D412yes—
305C3×C4#43512yes—
306C3×C4#46812yes—
307C3×C4#47012yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23#83012yes—
315C23C2×#21711yescharacteristic
316Q8—24no—
317Q8—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4#83012yes—
347C2×C4C22×C3⋊C811yescharacteristic
348C2×C4C2×#20811yescharacteristic
349C2×C4D4×Dic312yes—
350C2×C4#93511yescharacteristic
351C2×C4#93711yescharacteristic
352C2×C4#93912yes—
353C2×C4#94212yes—
354C2×C4#94812yes—
355C2×C4#94911yescharacteristic
356C2×C4#95111yescharacteristic
357C8—124no—
358C8—242no—
359C2×C3—42no—
360C2×C3#147711yescharacteristic
361C2×C3#147911yescharacteristic
362C2×C3#149511yescharacteristic
363C2×C3#149711yescharacteristic
364C2×C3#186911yescharacteristic
365C2×C3#189411yescharacteristic
366C2×C3#189711yescharacteristic
367C22—22no—
368C22—22no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22C2×#83011yescharacteristic
375C22#522011yescharacteristic
376C22#522511yescharacteristic
377C22#523211yescharacteristic
378C22#523411yescharacteristic
379C22#523611yescharacteristic
380C22#523611yescharacteristic
381C4—21no—
382C4—21no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—81no—
389C4—81no—
390C4D4×C3⋊C812yes—
391C4#501112yes—
392C4#504912yes—
393C4#505212yes—
394C3#907311yescharacteristic, complemented, Sylow
395C2—42no—
396C2#7161011yescharacteristic
397C2#7162111yescharacteristic
398C2#7163911yescharacteristic
399C2#7169711yescharacteristic
400C2#7171211yescharacteristic
401C2#7175311yescharacteristic
402C2#7182711yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×C22⋊Q8D3
hypocenterC3#9073
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#4535C2semidirect
5. #67619C2semidirect
36. C3×C22⋊Q8C8semidirect
147. C22⋊Q8C3⋊C8semidirect
394. C3#9073semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×#49252
C2×#45352
#715302
#676192
#90733