Finite Group Databasebeta

← #10690549

Subgroups — #10690549

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4×#208C211yesmaximal, characteristic
3C2×#4535C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67619C212yesmaximal, complemented
6#9077—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C22×#208—21no—
11C3×#391C411yescharacteristic
12C3×#389C411yescharacteristic
13C3×#387—21no—
14C3×#385—21no—
15C3×#380C411yescharacteristic
16C3×C4×C4⋊C4—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19C2×#830—21no—
20C2×C4⋊(C3⋊C8)C2212yes—
21C4×#208—24no—
22C2×#803—21no—
23C2×C4×C3⋊C8C2212yes—
24#4535—24no—
25#4524—21no—
26C2×C4×M16—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⋊Q8—22no—
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
55C2×#208—24no—
56C2×#208—24no—
57C2×#208—24no—
58C22×C3⋊C8—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65#830—42no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—22no—
68#803—24no—
69C4×C3⋊C8—22no—
70C4×C3⋊C8D414yes—
71C22×M16—61no—
72C2×C22⋊Q8Dic311yescharacteristic
73C2×C4×Q8C2×D311yescharacteristic
74C22×C42C2×D311yescharacteristic
75C2×C4⋊C8—32no—
76C2×C22⋊C8—61no—
77C4×M16—64no—
78C2×#97—61no—
79C2×C4×C8—32no—
80#391Dic311yescharacteristic
81#389Dic311yescharacteristic
82#387—21no—
83#385—21no—
84#380Dic311yescharacteristic
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×D311yescharacteristic
87C4×C22⋊C4—21no—
88#336—61no—
89Q8⋊C8—64no—
90C24×C3C22⋊C411yescharacteristic
91C2×C3×Q8—21no—
92C2×C3×Q8—22no—
93C2×C3×Q8—22no—
94C2×C3×Q8M1612yescomplemented
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4C22⋊C411yescharacteristic
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C3×C4⋊C4—24no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4M1612yescomplemented
125C3×C22⋊C4—24no—
126C3×C22⋊C4—42no—
127C3×C42—22no—
128C3×C42—21no—
129C3×C42—22no—
130C3×C42—21no—
131C3×C42—42no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C2×D412yes—
135C3×C42#4112yes—
136#208—216no—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—24no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—42no—
142C2×C3⋊C8—42no—
143C22×Q8C2×Dic311yescharacteristic
144C23×C4C2×Dic311yescharacteristic
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C2×M16—64no—
148C2×M16—64no—
149C2×M16—64no—
150C22×C8—62no—
151C22×C8—62no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic311yescharacteristic
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C4⋊C4C2×Dic311yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2×Dic311yescharacteristic
169C2×C42#6711yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42C22×D311yescharacteristic
172C4⋊C8—62no—
173C4⋊C8—62no—
174C22⋊C8—122no—
175#97—64no—
176C4×C8—34no—
177C4×C8—62no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3#10412yes—
190C23×C3C2×C22⋊C411yescharacteristic
191C3×Q8—24no—
192C3×Q8—42no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4#10412yes—
222C2×C3×C4C4×D412yes—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12112yes—
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4C2×M1611yescharacteristic
227C2×C3×C4C2×M1611yescharacteristic
228C2×C3×C4#13112yes—
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13712yes—
231C2×C3×C4#13711yescharacteristic
232C3⋊C8—28no—
233C3⋊C8—82no—
234C24#21711yescharacteristic
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8#20812yescomplemented
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4#21711yescharacteristic
256C22×C4#21711yescharacteristic
257C22×C4#21711yescharacteristic
258C22×C4C22×Dic311yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—616no—
262C2×C8—64no—
263C2×C8—64no—
264C2×C8—64no—
265C2×C8—122no—
266C2×C8—122no—
267C2×C8—122no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4#20812yescomplemented
276C22⋊C4—24no—
277C22⋊C4—42no—
278C42—22no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42D3×D412yes—
286C42#23712yes—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3#38611yescharacteristic
295C22×C3#40711yescharacteristic
296C22×C3#40811yescharacteristic
297C22×C3#41711yescharacteristic
298C22×C3#41911yescharacteristic
299C22×C3C2×#10411yescharacteristic
300C22×C3C2×#10411yescharacteristic
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—41no—
307C3×C4—42no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4#43512yes—
311C3×C4#43612yes—
312C3×C4#46812yes—
313C3×C4#47012yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#83712yes—
321C23C2×#21711yescharacteristic
322Q8—24no—
323Q8—42no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4#83712yes—
353C2×C4C2×#20811yescharacteristic
354C2×C4C2×#20811yescharacteristic
355C2×C4D4×Dic312yes—
356C2×C4#93511yescharacteristic
357C2×C4#93711yescharacteristic
358C2×C4#93912yes—
359C2×C4#94212yes—
360C2×C4#94812yes—
361C2×C4#94911yescharacteristic
362C2×C4#95111yescharacteristic
363C8—68no—
364C8—242no—
365C2×C3—42no—
366C2×C3#148311yescharacteristic
367C2×C3#148511yescharacteristic
368C2×C3#148911yescharacteristic
369C2×C3#149111yescharacteristic
370C2×C3#187011yescharacteristic
371C2×C3#189411yescharacteristic
372C2×C3#189611yescharacteristic
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22#520711yescharacteristic
381C22#522011yescharacteristic
382C22#522511yescharacteristic
383C22#523211yescharacteristic
384C22#523411yescharacteristic
385C22C2×#83711yescharacteristic
386C22C2×#83711yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4#501112yes—
397C4#501212yes—
398C4#504912yes—
399C4#505212yes—
400C3#907711yescharacteristic, complemented, Sylow
401C2—42no—
402C2#7161111yescharacteristic
403C2#7162311yescharacteristic
404C2#7164111yescharacteristic
405C2#7169711yescharacteristic
406C2#7171111yescharacteristic
407C2#7175111yescharacteristic
408C2#7182511yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×C22⋊Q8D3
hypocenterC3#9077
LCSG ▷ C22×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
94. C2×C3×Q8M16semidirect
124. C3×C4⋊C4M16semidirect
238. C2×Q8#208semidirect
275. C4⋊C4#208semidirect
400. C3#9077semidirect

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×C4×#2082
C2×#45352
#715302
#676192
#90773