Finite Group Databasebeta

← #16778462

Subgroups — #16778462

Order 3328.

506 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130727C211yesmaximal, characteristic
2#13130562C211yesmaximal, characteristic
3#13129656C212yesmaximal, complemented
4#13129654C212yesmaximal, complemented
5C22⋊Q8×Dic13C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C411yescharacteristic
7C2×Q8×Dic13C411yescharacteristic
8C22×C4×Dic13C2211yescharacteristic
9C2×C22⋊#1057—21no—
10C2×C22⋊#1057—21no—
11C2×#88110—21no—
12#11718891C2211yescharacteristic
13#11718873C411yescharacteristic
14#11718646C2211yescharacteristic
15#11718642—21no—
16C4⋊C4×Dic13—21no—
17C4⋊C4×Dic13C411yescharacteristic
18#11718611—21no—
19C22⋊C4×Dic13—21no—
20#11718290—21no—
21#11718290C2212yes—
22#11718290C2212yes—
23C22×Q8×C13C2×C411yescharacteristic
24C23×C4×C13C2×C411yescharacteristic
25C23×Dic13D411yescharacteristic
26C23×Dic13D411yescharacteristic
27C22×#1057—22no—
28C22×#1057—22no—
29C13×C22⋊Q8—22no—
30C13×C22⋊Q8—22no—
31C2×C13×C4⋊C4—21no—
32C2×C13×C4⋊C4C2×C411yescharacteristic
33C2×C13×C22⋊C4—21no—
34C2×C22⋊Dic13—21no—
35Q8×Dic13—42no—
36C2×C4⋊Dic13—21no—
37C2×C4⋊Dic13—21no—
38C2×C4⋊Dic13C2×C411yescharacteristic
39C2×C4⋊Dic13C2×C411yescharacteristic
40C2×C4×Dic13—21no—
41C2×C4×Dic13—21no—
42C2×C4×Dic13—42no—
43C2×C4×Dic13C2×C411yescharacteristic
44C2×C4×Dic13D411yescharacteristic
45C2×C4×Dic13D411yescharacteristic
46C2×C4×Dic13C2311yescharacteristic
47C22⋊#1057—24no—
48C22⋊#1057—42no—
49#88110—42no—
50#88071—21no—
51#88071—21no—
52#88071—21no—
53#88071—21no—
54#88071—21no—
55#7623—131nomaximal, Sylow
56C24×C13C22⋊C411yescharacteristic
57C2×Q8×C13—21no—
58C2×Q8×C13—41no—
59C2×Q8×C13—41no—
60C2×Q8×C13M1612yes—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—21no—
65C22×C4×C13—41no—
66C22×C4×C13—41no—
67C22×C4×C13C22⋊C411yescharacteristic
68C22×C4×C13C22×C411yescharacteristic
69C22×Dic13—22no—
70C22×Dic13—22no—
71C22×Dic13—22no—
72C22×Dic13—22no—
73C22×Dic13—21no—
74C22×Dic13—21no—
75C22×Dic13—41no—
76C22×Dic13—41no—
77C22×Dic13C22⋊C411yescharacteristic
78C22×Dic13C22⋊C411yescharacteristic
79C22×Dic13C2×D411yescharacteristic
80C22×Dic13C2×D411yescharacteristic
81C2×#1057—24no—
82C2×#1057—42no—
83C2×#1057—42no—
84C2×#1057—42no—
85C2×#1057—42no—
86C2×#1057—42no—
87C13×C4⋊C4—24no—
88C13×C4⋊C4—21no—
89C13×C4⋊C4M1612yes—
90C13×C22⋊C4—24no—
91C22⋊Dic13—42no—
92C4⋊Dic13—42no—
93C4⋊Dic13—42no—
94C4⋊Dic13—41no—
95C4⋊Dic13—41no—
96C4×Dic13—22no—
97C4×Dic13—22no—
98C4×Dic13—41no—
99C4×Dic13—41no—
100C4×Dic13—42no—
101C4×Dic13—42no—
102C4×C22⋊Q8—131no—
103#1817—131no—
104#1749—131no—
105#1272—132no—
106#1271—132no—
107C23×C13—21no—
108C23×C13—21no—
109C23×C13—21no—
110C23×C13—21no—
111C23×C13—21no—
112C23×C13—21no—
113C23×C13#10412yes—
114C23×C13C2×C22⋊C411yescharacteristic
115Q8×C13—42no—
116Q8×C13—42no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—22no—
120C2×C4×C13—22no—
121C2×C4×C13—21no—
122C2×C4×C13—21no—
123C2×C4×C13—21no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—41no—
127C2×C4×C13—41no—
128C2×C4×C13—41no—
129C2×C4×C13—41no—
130C2×C4×C13C2×M1611yescharacteristic
131C2×C4×C13C2×M1611yescharacteristic
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—21no—
135C2×Dic13—21no—
136C2×Dic13—21no—
137C2×Dic13—21no—
138C2×Dic13—21no—
139C2×Dic13—21no—
140C2×Dic13—21no—
141C2×Dic13—44no—
142C2×Dic13—44no—
143C2×Dic13—41no—
144C2×Dic13—41no—
145C2×Dic13—41no—
146C2×Dic13—41no—
147C2×Dic13—41no—
148C2×Dic13—41no—
149C2×Dic13#9912yes—
150C2×Dic13#13512yes—
151C2×Dic13#13712yes—
152#1057—44no—
153#1057—82no—
154C2×C22⋊Q8#26111yescharacteristic
155C2×C4×Q8—131no—
156C22×C42—131no—
157C2×C22⋊C8—261no—
158C2×C22⋊C8—261no—
159C2×#97—261no—
160#391—131no—
161#389—131no—
162#387—261no—
163#385—261no—
164#380—131no—
165C4×C4⋊C4—131no—
166C4×C4⋊C4—261no—
167C4×C22⋊C4—261no—
168#336—132no—
169#336—132no—
170#336—261no—
171C22×C13—22no—
172C22×C13—22no—
173C22×C13—22no—
174C22×C13—41no—
175C22×C13—41no—
176C22×C13—41no—
177C22×C13—41no—
178C22×C13#40711yescharacteristic
179C22×C13C2×#9911yescharacteristic
180C22×C13#41311yescharacteristic
181C22×C13#41611yescharacteristic
182C22×C13#41911yescharacteristic
183C22×C13C2×#10411yescharacteristic
184C22×C13#42111yescharacteristic
185C4×C13—21no—
186C4×C13—21no—
187C4×C13—41no—
188C4×C13—42no—
189C4×C13—42no—
190Dic13—22no—
191Dic13—22no—
192Dic13—22no—
193Dic13—41no—
194Dic13—81no—
195Dic13—81no—
196C22×Q8C2×#26111yescharacteristic
197C23×C4—131no—
198C23×C4—131no—
199C23×C4C2×#26111yescharacteristic
200C22×C8—262no—
201C22×C8—262no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C4×Q8—522no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—131no—
207C2×C4⋊C4—131no—
208C2×C4⋊C4—261no—
209C2×C4⋊C4—261no—
210C2×C4⋊C4C2×#26111yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—261no—
213C2×C42—131no—
214C2×C42—131no—
215C2×C42—131no—
216C2×C42—131no—
217C2×C42—261no—
218C2×C42—261no—
219C2×C42—522no—
220C22⋊C8—264no—
221C22⋊C8—522no—
222#97—522no—
223#95—261no—
224#95—261no—
225#95—261no—
226#95—261no—
227#95—261no—
228C2×C13—42no—
229C2×C13#146511yescharacteristic
230C2×C13#148811yescharacteristic
231C2×C13#149111yescharacteristic
232C2×C13#150111yescharacteristic
233C2×C13#151111yescharacteristic
234C2×C13#153211yescharacteristic
235C2×C13#155411yescharacteristic
236C24C22⋊#26111yescharacteristic
237C2×Q8—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#612212yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—131no—
248C22×C4—131no—
249C22×C4—131no—
250C22×C4—131no—
251C22×C4—262no—
252C22×C4—262no—
253C22×C4—262no—
254C22×C4—262no—
255C22×C4—261no—
256C22×C4—261no—
257C22×C4—521no—
258C22×C4—521no—
259C22×C4C22⋊#26111yescharacteristic
260C22×C4C22×#26111yescharacteristic
261C2×C8—264no—
262C2×C8—522no—
263C2×C8—522no—
264C2×C8—522no—
265C2×C8—522no—
266C2×C8—522no—
267C4⋊C4—24no—
268C4⋊C4—21no—
269C4⋊C4—521no—
270C4⋊C4—521no—
271C4⋊C4—522no—
272C4⋊C4—522no—
273C4⋊C4#612212yes—
274C22⋊C4—24no—
275C22⋊C4—522no—
276C42—262no—
277C42—262no—
278C42—521no—
279C42—521no—
280C42—522no—
281C42—522no—
282C13#762311yescharacteristic, complemented, Sylow
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#8811812yes—
290C23C2×C22⋊#26111yescharacteristic
291Q8—42no—
292Q8—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—132no—
307C2×C4—132no—
308C2×C4—132no—
309C2×C4—261no—
310C2×C4—261no—
311C2×C4—261no—
312C2×C4—261no—
313C2×C4—261no—
314C2×C4—261no—
315C2×C4—261no—
316C2×C4—261no—
317C2×C4—261no—
318C2×C4—524no—
319C2×C4—524no—
320C2×C4—521no—
321C2×C4—521no—
322C2×C4—521no—
323C2×C4—521no—
324C2×C4—521no—
325C2×C4—521no—
326C2×C4C2×#612211yescharacteristic
327C2×C4C2×#612211yescharacteristic
328C8—524no—
329C8—1042no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22D13⋊#9911yescharacteristic
338C22#1171914811yescharacteristic
339C22#1171915311yescharacteristic
340C22C2×#8811811yescharacteristic
341C22#1171916511yescharacteristic
342C22#1171916611yescharacteristic
343C22#1171916911yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—262no—
350C4—262no—
351C4—262no—
352C4—521no—
353C4—1041no—
354C4—1041no—
355C2—42no—
356C2#1313009211yescharacteristic
357C2#1313016711yescharacteristic
358C2#1313017811yescharacteristic
359C2#1313021411yescharacteristic
360C2#1313024511yescharacteristic
361C2#1313031411yescharacteristic
362C2#1313036211yescharacteristic
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#7623
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13129656C2semidirect
4. #13129654C2semidirect
282. C13#7623semidirect

Maximal subgroups

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

subgroupindex
#131307272
#131305622
#131296562
#131296542
C22⋊Q8×Dic132
#762313