Finite Group Databasebeta

← #15782072

Subgroups — #15782072

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718818C211yesmaximal, characteristic
2C2×#11718081C211yesmaximal, characteristic, complemented
3C2×#11718154C211yesmaximal, characteristic, complemented
4C2×#11718099C211yesmaximal, characteristic, complemented
5#13114862C211yesmaximal, characteristic, complemented
6#13112928C211yesmaximal, characteristic, complemented
7C2×C13×#357C211yesmaximal, characteristic, complemented
8C22×C13×D8C2211yescharacteristic
9C2×#88192—21no—
10C22×C8×D13C2211yescharacteristic
11C22×C13×C16C2211yescharacteristic
12C2×C13×#107C2211yescharacteristic
13C2×#88072—21no—
14#11718818—22no—
15#11718818—22no—
16C2×#88058C2211yescharacteristic
17C2×#88058C2211yescharacteristic
18C22×#6090C2211yescharacteristic
19#11718648—21no—
20C13×#357—22no—
21#11718154—22no—
22#11718121—21no—
23#11718099—22no—
24#11718081—22no—
25C22×D4×C13—21no—
26C22×C22⋊D13—41no—
27C22×C4×D13D411yescharacteristic
28C2×C13×D8—21no—
29C2×C13×D8—21no—
30C2×C13×D8—21no—
31C2×C13×D8—21no—
32C2×C13×D8—21no—
33C2×C13×D8D412yes—
34C22×C8×C13C2311yescharacteristic
35C2×C13×C4⋊C4—21no—
36C2×C22⋊Dic13—41no—
37#88192—24no—
38#88192—42no—
39C2×#6103—41no—
40C2×C4⋊Dic13—21no—
41C2×C4⋊Dic13—21no—
42C22×#1055D411yescharacteristic
43C2×C8×D13—21no—
44C2×C8×D13—21no—
45C2×C8×D13—21no—
46C2×C8×D13—21no—
47C2×C8×D13—21no—
48C2×C8×D13—21no—
49C2×C13×C16—21no—
50C2×C13×C16—21no—
51C2×C13×C16—21no—
52C13×#107—21no—
53C13×#107—21no—
54#88072—24no—
55#88071—41no—
56#88058—21no—
57#88058—21no—
58#88058—21no—
59#88058D412yes—
60C2×#6090—21no—
61C2×#6090—21no—
62C2×#6090D412yescomplemented
63#13380—131nomaximal, Sylow
64C24×C13—41no—
65C23×D13D811yescharacteristic
66C2×D4×C13—22no—
67C2×D4×C13—41no—
68C2×D4×C13—41no—
69C2×D4×C13—41no—
70C2×D4×C13—41no—
71C2×D4×C13—41no—
72C22×C4×C13—41no—
73C22×C4×C13C2×D411yescharacteristic
74C2×C22⋊D13—44no—
75C2×C22⋊D13—82no—
76C2×C22⋊D13—82no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C22×Dic13D811yescharacteristic
80C2×C4×D13—21no—
81C2×C4×D13—21no—
82C2×C4×D13—21no—
83C2×C4×D13—21no—
84C2×C4×D13—21no—
85C2×C4×D13—21no—
86C13×D8—22no—
87C13×D8—22no—
88C13×D8—41no—
89C13×D8—41no—
90C2×C8×C13C2×D411yescharacteristic
91C2×C8×C13C2×D411yescharacteristic
92C2×C8×C13C2×D411yescharacteristic
93C2×C8×C13#4111yescharacteristic
94C2×C8×C13#4111yescharacteristic
95C2×C8×C13#4111yescharacteristic
96C13×C4⋊C4—41no—
97C13×C4⋊C4—41no—
98C22⋊Dic13—44no—
99#6103—82no—
100C4⋊Dic13—22no—
101C4⋊Dic13—41no—
102C4⋊Dic13—41no—
103C4⋊Dic13—41no—
104C2×#1055—21no—
105C2×#1055—21no—
106C2×#1055—21no—
107C8×D13—41no—
108C8×D13—41no—
109C8×D13—41no—
110C8×D13—41no—
111C13×C16—41no—
112#6090—22no—
113C2×#466—131no—
114C2×#366—131no—
115C2×#357—131no—
116C2×#357D1311yescharacteristic, complemented
117C2×C22⋊C16—131no—
118#1808—131no—
119#1380—131no—
120C23×C13—42no—
121C23×C13—81no—
122C23×C13—81no—
123C23×C13—81no—
124C23×C13—81no—
125C23×C13—81no—
126C23×C13—81no—
127C23×C13C2×D811yescharacteristic
128C22×D13—21no—
129C22×D13—21no—
130C22×D13—21no—
131C22×D13—21no—
132C22×D13—21no—
133C22×D13—21no—
134C22×D13D1612yes—
135D4×C13—42no—
136D4×C13—42no—
137D4×C13—81no—
138D4×C13—81no—
139C2×C4×C13—81no—
140C2×C4×C13—81no—
141C2×C4×C13—81no—
142C2×C4×C13#12011yescharacteristic
143C2×C4×C13#12111yescharacteristic
144C2×C4×C13#12311yescharacteristic
145C2×C4×C13#13511yescharacteristic
146C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C13#13711yescharacteristic
148C22⋊D13—88no—
149C2×Dic13—21no—
150C2×Dic13—21no—
151C2×Dic13—21no—
152C2×Dic13—42no—
153C2×Dic13—81no—
154C2×Dic13—81no—
155C2×Dic13—81no—
156C2×Dic13—81no—
157C2×Dic13—81no—
158C4×D13—41no—
159C4×D13—41no—
160C4×D13—41no—
161C4×D13—41no—
162C8×C13#12111yescharacteristic
163C8×C13#12111yescharacteristic
164C8×C13#12311yescharacteristic
165C8×C13#12411yescharacteristic
166#1055—41no—
167C22×D8C2×D1311yescharacteristic
168C23×C8—131no—
169C2×#121—261no—
170C22×C16—131no—
171C22×C16C2×D1311yescharacteristic
172#466—262no—
173#466—262no—
174C2×#107—131no—
175C2×#107—131no—
176C2×#107C2×D1311yescharacteristic
177C2×#102—261no—
178#380—261no—
179#366—262no—
180#357—22no—
181#357—262no—
182C22⋊C16—262no—
183#340—261no—
184C22×C13—82no—
185C22×C13—82no—
186C22×C13—82no—
187C22×C13—161no—
188C22×C13—161no—
189C22×C13—161no—
190C22×C13—161no—
191C22×C13#44711yescharacteristic
192C22×C13#46611yescharacteristic
193C22×C13#48011yescharacteristic
194C22×C13C2×D1611yescharacteristic
195C22×C13#50811yescharacteristic
196C22×C13#50911yescharacteristic
197C22×C13#51011yescharacteristic
198C2×D13—22no—
199C2×D13—22no—
200C2×D13—22no—
201C2×D13—41no—
202C2×D13—41no—
203C2×D13—41no—
204C2×D13—41no—
205C4×C13—161no—
206C4×C13#39411yescharacteristic
207C4×C13#45211yescharacteristic
208C4×C13#46811yescharacteristic
209C4×C13#48211yescharacteristic
210Dic13—41no—
211Dic13—82no—
212Dic13—161no—
213C22×D4—21no—
214C22×D4—521no—
215C23×C4—131no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—21no—
221C2×D8C22⋊D1312yes—
222C22×C8—131no—
223C22×C8—261no—
224C22×C8—261no—
225C22×C8—261no—
226C22×C8—261no—
227C22×C8—261no—
228C22×C8—261no—
229C22×C8C22×D1311yescharacteristic
230#121—264no—
231#121—522no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—261no—
234C2×C4⋊C4—261no—
235C2×C22⋊C4—521no—
236C2×C22⋊C4—521no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—132no—
241C2×C16—261no—
242C2×C16—261no—
243#107—21no—
244#107—21no—
245#107—132no—
246#107—261no—
247#107—261no—
248#107—261no—
249#102—264no—
250#95—521no—
251C2×C13—162no—
252C2×C13#203311yescharacteristic
253C2×C13#218411yescharacteristic
254C2×C13#218911yescharacteristic
255C2×C13#221511yescharacteristic
256C2×C13#222011yescharacteristic
257C2×C13#223211yescharacteristic
258C2×C13#223411yescharacteristic
259D13—42no—
260C24—41no—
261C24—131no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—524no—
269C2×D4—1042no—
270C2×D4—1042no—
271C22×C4—41no—
272C22×C4—131no—
273C22×C4—261no—
274C22×C4—261no—
275C22×C4—261no—
276C22×C4—261no—
277C22×C4—261no—
278C22×C4—261no—
279C22×C4—521no—
280C22×C4—521no—
281C22×C4D4×D1311yescharacteristic
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286C2×C8—261no—
287C2×C8—261no—
288C2×C8—261no—
289C2×C8—521no—
290C2×C8—521no—
291C2×C8—521no—
292C2×C8—521no—
293C2×C8#612711yescharacteristic
294C2×C8#612711yescharacteristic
295C2×C8D4×D1311yescharacteristic
296C2×C8D4×D1311yescharacteristic
297C2×C8#612911yescharacteristic
298C2×C8C2×C22⋊D1311yescharacteristic
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—262no—
302C4⋊C4—521no—
303C4⋊C4—521no—
304C4⋊C4—521no—
305C22⋊C4—524no—
306C22⋊C4—1042no—
307C16—41no—
308C16—262no—
309C13#1338011yescharacteristic, complemented, Sylow
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—132no—
318C23—261no—
319C23—261no—
320C23—261no—
321C23—261no—
322C23—261no—
323C23—261no—
324C23D8×D1311yescharacteristic
325D4—42no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—1048no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—261no—
334C2×C4—261no—
335C2×C4—261no—
336C2×C4—521no—
337C2×C4—521no—
338C2×C4—521no—
339C2×C4—521no—
340C2×C4—522no—
341C2×C4—1041no—
342C2×C4—1041no—
343C2×C4—1041no—
344C2×C4—1041no—
345C2×C4—1041no—
346C2×C4#8813711yescharacteristic
347C2×C4#8813811yescharacteristic
348C2×C4#8816511yescharacteristic
349C2×C4#8816911yescharacteristic
350C2×C4#8817011yescharacteristic
351C2×C4#8819111yescharacteristic
352C8—521no—
353C8#8813811yescharacteristic
354C8#8813911yescharacteristic
355C8#8818311yescharacteristic
356C8#8819311yescharacteristic
357C22—82no—
358C22—82no—
359C22—82no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—262no—
365C22—262no—
366C22—262no—
367C22—521no—
368C22—521no—
369C22—521no—
370C22—521no—
371C22#1171843511yescharacteristic
372C22#1171843611yescharacteristic
373C22D13×D1611yescharacteristic
374C22#1171857211yescharacteristic
375C22#1171857611yescharacteristic
376C22#1171857711yescharacteristic
377C22#1171881711yescharacteristic
378C4—161no—
379C4—521no—
380C4—1042no—
381C4—2081no—
382C4#1171844011yescharacteristic
383C4#1171844111yescharacteristic
384C4#1171862011yescharacteristic
385C4#1171883411yescharacteristic
386C2—162no—
387C2—522no—
388C2#1311601611yescharacteristic
389C2#1311674611yescharacteristic
390C2#1311677311yescharacteristic
391C2#1311692111yescharacteristic
392C2#1311694811yescharacteristic
393C2#1311701511yescharacteristic
394C2#1311702611yescharacteristic
subgroupG/H
centerC23D8×D13
commutator, cosocleC22×C8×C13C23
FrattiniC22×C8C22×D13
FittingC2×C13×#357C2
socleC23×C13C2×D8
prime radicalC2×#11718818C2
hypercenterC2×#357D13
hypocenterC13#13380
LCSG ▷ C22×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#11718081C2semidirect
3. C2×#11718154C2semidirect
4. C2×#11718099C2semidirect
5. #13114862C2semidirect
6. #13112928C2semidirect
7. C2×C13×#357C2semidirect
62. C2×#6090D4semidirect
116. C2×#357D13semidirect
309. C13#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#117188182
C2×#117180812
C2×#117181542
C2×#117180992
#131148622
#131129282
C2×C13×#3572
#1338013