Finite Group Databasebeta

← #16782727

Subgroups — #16782727

Order 3328.

604 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719130C211yesmaximal, characteristic
2C4×C22⋊#1057C211yesmaximal, characteristic
3C2×Q8⋊#1057C212yesmaximal, complemented
4#13129672C212yesmaximal, complemented
5C22⋊Q8×Dic13C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C411yescharacteristic
7C2×Q8×Dic13C2211yescharacteristic
8C22×C4×Dic13C2211yescharacteristic
9C2×C22⋊#1057—21no—
10C2×C22⋊#1057—21no—
11#11719130—24no—
12C2×#88110—21no—
13C2×C4⋊#1057C2212yes—
14C2×C4×#1057C2212yes—
15#11718891C411yescharacteristic
16#11718873C411yescharacteristic
17#11718646C411yescharacteristic
18#11718642—21no—
19C4⋊C4×Dic13—21no—
20C4⋊C4×Dic13C2211yescharacteristic
21#11718611—21no—
22C22⋊C4×Dic13—21no—
23Q8⋊#1057—24no—
24#11718290—21no—
25C22×Q8×C13C2×C411yescharacteristic
26C23×C4×C13C2×C411yescharacteristic
27C23×Dic13D411yescharacteristic
28C23×Dic13D411yescharacteristic
29C22×#1057—22no—
30C22×#1057—22no—
31C13×C22⋊Q8—22no—
32C13×C22⋊Q8—22no—
33C2×C13×C4⋊C4—21no—
34C2×C13×C4⋊C4C2×C411yescharacteristic
35C2×C13×C22⋊C4—21no—
36C2×C22⋊Dic13—21no—
37Q8×Dic13—22no—
38Q8×Dic13—22no—
39C2×C4⋊Dic13—21no—
40C2×C4⋊Dic13—21no—
41C2×C4⋊Dic13C2×C411yescharacteristic
42C2×C4⋊Dic13C2×C411yescharacteristic
43C2×C4×Dic13—22no—
44C2×C4×Dic13—22no—
45C2×C4×Dic13—21no—
46C2×C4×Dic13—21no—
47C2×C4×Dic13C2×C411yescharacteristic
48C2×C4×Dic13D411yescharacteristic
49C2×C4×Dic13D411yescharacteristic
50C2×C4×Dic13C2311yescharacteristic
51C22⋊#1057—24no—
52C22⋊#1057—24no—
53#88110—24no—
54C4⋊#1057—22no—
55C4⋊#1057D414yes—
56C4×#1057—22no—
57C4×#1057—22no—
58#88071—21no—
59#88071—21no—
60#88071—21no—
61#88071—21no—
62#88071—21no—
63#8941—131nomaximal, Sylow
64C24×C13C22⋊C411yescharacteristic
65C2×Q8×C13—21no—
66C2×Q8×C13—22no—
67C2×Q8×C13—22no—
68C2×Q8×C13M1612yes—
69C22×C4×C13—21no—
70C22×C4×C13—21no—
71C22×C4×C13—22no—
72C22×C4×C13—21no—
73C22×C4×C13—21no—
74C22×C4×C13—22no—
75C22×C4×C13C22⋊C411yescharacteristic
76C22×C4×C13C22×C411yescharacteristic
77C22×Dic13—22no—
78C22×Dic13—21no—
79C22×Dic13—22no—
80C22×Dic13—21no—
81C22×Dic13—22no—
82C22×Dic13—22no—
83C22×Dic13—21no—
84C22×Dic13—21no—
85C22×Dic13—41no—
86C22×Dic13C22⋊C411yescharacteristic
87C22×Dic13C22⋊C411yescharacteristic
88C22×Dic13C2×D411yescharacteristic
89C22×Dic13C2×D411yescharacteristic
90C2×#1057—24no—
91C2×#1057—24no—
92C2×#1057—42no—
93C2×#1057—42no—
94C2×#1057—42no—
95C2×#1057—42no—
96C13×C4⋊C4—24no—
97C13×C4⋊C4—21no—
98C13×C4⋊C4M1612yes—
99C13×C22⋊C4—24no—
100C22⋊Dic13—42no—
101C4⋊Dic13—22no—
102C4⋊Dic13—22no—
103C4⋊Dic13—42no—
104C4⋊Dic13—42no—
105C4×Dic13—22no—
106C4×Dic13—21no—
107C4×Dic13—22no—
108C4×Dic13—21no—
109C4×Dic13—41no—
110C4×Dic13—41no—
111C4×Dic13—42no—
112C4×Dic13C2×D412yes—
113C4×Dic13#4112yes—
114C2×#432—131no—
115C4×C22⋊Q8—131no—
116C4×C22⋊C8—131no—
117C2×Q8⋊C8—132no—
118#1279—132no—
119C23×C13—21no—
120C23×C13—21no—
121C23×C13—21no—
122C23×C13—21no—
123C23×C13—21no—
124C23×C13—21no—
125C23×C13#10412yes—
126C23×C13C2×C22⋊C411yescharacteristic
127Q8×C13—24no—
128Q8×C13—42no—
129C2×C4×C13—21no—
130C2×C4×C13—22no—
131C2×C4×C13—22no—
132C2×C4×C13—21no—
133C2×C4×C13—21no—
134C2×C4×C13—41no—
135C2×C4×C13—41no—
136C2×C4×C13—41no—
137C2×C4×C13—41no—
138C2×C4×C13—42no—
139C2×C4×C13—42no—
140C2×C4×C13C4×D412yes—
141C2×C4×C13C2×M1611yescharacteristic
142C2×C4×C13C2×M1611yescharacteristic
143C2×C4×C13#13112yes—
144C2×Dic13—21no—
145C2×Dic13—21no—
146C2×Dic13—21no—
147C2×Dic13—21no—
148C2×Dic13—21no—
149C2×Dic13—21no—
150C2×Dic13—21no—
151C2×Dic13—21no—
152C2×Dic13—44no—
153C2×Dic13—42no—
154C2×Dic13—42no—
155C2×Dic13—41no—
156C2×Dic13—41no—
157C2×Dic13—41no—
158C2×Dic13—41no—
159C2×Dic13—41no—
160C2×Dic13—41no—
161C2×Dic13#12011yescharacteristic
162C2×Dic13#12112yes—
163C2×Dic13#12311yescharacteristic
164C2×Dic13#13512yes—
165C2×Dic13#13712yes—
166#1057—44no—
167#1057—44no—
168C2×C22⋊Q8#26111yescharacteristic
169C2×C4×Q8—131no—
170C22×C42—131no—
171#432—264no—
172C2×C4⋊C8—132no—
173C2×C22⋊C8—261no—
174C2×C22⋊C8—261no—
175C2×#97—261no—
176C2×C4×C8—132no—
177#391—131no—
178#389—131no—
179#387—261no—
180#385—261no—
181#380—131no—
182C4×C4⋊C4—131no—
183C4×C4⋊C4—261no—
184C4×C22⋊C4—261no—
185#336—261no—
186Q8⋊C8—264no—
187C22×C13—22no—
188C22×C13—22no—
189C22×C13—22no—
190C22×C13—41no—
191C22×C13—41no—
192C22×C13—41no—
193C22×C13—41no—
194C22×C13#38611yescharacteristic
195C22×C13#40711yescharacteristic
196C22×C13#40811yescharacteristic
197C22×C13#41611yescharacteristic
198C22×C13#41911yescharacteristic
199C22×C13C2×#10411yescharacteristic
200C22×C13#42111yescharacteristic
201C4×C13—41no—
202C4×C13—42no—
203C4×C13—42no—
204C4×C13#43512yes—
205C4×C13#43612yes—
206Dic13—22no—
207Dic13—41no—
208Dic13—41no—
209Dic13—81no—
210Dic13—81no—
211Dic13#45212yes—
212Dic13#48512yes—
213C22×Q8C2×#26111yescharacteristic
214C23×C4—131no—
215C23×C4—131no—
216C23×C4C2×#26111yescharacteristic
217C22×C8—262no—
218C22×C8—262no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C4×Q8—262no—
222C4×Q8—262no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—131no—
225C2×C4⋊C4—131no—
226C2×C4⋊C4—261no—
227C2×C4⋊C4—261no—
228C2×C4⋊C4C2×#26111yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—261no—
231C2×C42—131no—
232C2×C42—131no—
233C2×C42—131no—
234C2×C42—131no—
235C2×C42—262no—
236C2×C42—262no—
237C2×C42—261no—
238C2×C42—261no—
239C4⋊C8—134no—
240C4⋊C8—262no—
241C22⋊C8—264no—
242C22⋊C8—264no—
243#97—264no—
244C4×C8—262no—
245C4×C8—262no—
246#95—261no—
247#95—261no—
248#95—261no—
249#95—261no—
250#95—261no—
251C2×C13—42no—
252C2×C13#148811yescharacteristic
253C2×C13#149111yescharacteristic
254C2×C13#149411yescharacteristic
255C2×C13#149711yescharacteristic
256C2×C13#187011yescharacteristic
257C2×C13#187111yescharacteristic
258C2×C13#187311yescharacteristic
259C24C22⋊#26111yescharacteristic
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8#612212yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—131no—
271C22×C4—131no—
272C22×C4—131no—
273C22×C4—131no—
274C22×C4—261no—
275C22×C4—261no—
276C22×C4—262no—
277C22×C4—262no—
278C22×C4—262no—
279C22×C4—262no—
280C22×C4—261no—
281C22×C4—261no—
282C22×C4—521no—
283C22×C4C22⋊#26111yescharacteristic
284C22×C4C22×#26111yescharacteristic
285C2×C8—264no—
286C2×C8—264no—
287C2×C8—522no—
288C2×C8—522no—
289C2×C8—522no—
290C2×C8—522no—
291C4⋊C4—24no—
292C4⋊C4—21no—
293C4⋊C4—262no—
294C4⋊C4—262no—
295C4⋊C4—522no—
296C4⋊C4—522no—
297C4⋊C4#612212yes—
298C22⋊C4—24no—
299C22⋊C4—522no—
300C42—132no—
301C42—132no—
302C42—261no—
303C42—261no—
304C42—262no—
305C42—262no—
306C42—521no—
307C42—521no—
308C42—522no—
309C13#894111yescharacteristic, complemented, Sylow
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23#8811812yes—
317C23C2×C22⋊#26111yescharacteristic
318Q8—24no—
319Q8—42no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—131no—
332C2×C4—131no—
333C2×C4—132no—
334C2×C4—132no—
335C2×C4—132no—
336C2×C4—261no—
337C2×C4—261no—
338C2×C4—261no—
339C2×C4—261no—
340C2×C4—261no—
341C2×C4—261no—
342C2×C4—261no—
343C2×C4—261no—
344C2×C4—522no—
345C2×C4—522no—
346C2×C4—524no—
347C2×C4—521no—
348C2×C4—521no—
349C2×C4—521no—
350C2×C4—521no—
351C2×C4—521no—
352C2×C4—521no—
353C2×C4#8823812yes—
354C2×C4D4×#26112yes—
355C2×C4C2×#612211yescharacteristic
356C2×C4C2×#612211yescharacteristic
357C8—524no—
358C8—524no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22#1171914611yescharacteristic
367C22#1171915011yescharacteristic
368C22#1171915311yescharacteristic
369C22C2×#8811811yescharacteristic
370C22#1171916511yescharacteristic
371C22#1171916611yescharacteristic
372C22#1171916911yescharacteristic
373C4—41no—
374C4—42no—
375C4—42no—
376C4—132no—
377C4—132no—
378C4—262no—
379C4—521no—
380C4—521no—
381C4—1041no—
382C4—1041no—
383C4#1171914412yes—
384C4#1171914512yes—
385C2—42no—
386C2#1313016711yescharacteristic
387C2#1313017811yescharacteristic
388C2#1313018911yescharacteristic
389C2#1313019811yescharacteristic
390C2#1313081311yescharacteristic
391C2#1313081711yescharacteristic
392C2#1313082211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×C22⋊Q8C4
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC2×C22⋊Q8#261
hypocenterC13#8941
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#1057C2semidirect
4. #13129672C2semidirect
309. C13#8941semidirect

Maximal subgroups

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

subgroupindex
C2×#117191302
C4×C22⋊#10572
C2×Q8⋊#10572
#131296722
C22⋊Q8×Dic132
#894113