Finite Group Databasebeta

← #16857607

Subgroups — #16857607

Order 3968.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521708C211yesmaximal, characteristic, complemented
2#14521406C211yesmaximal, characteristic, complemented
3#14521390C211yesmaximal, characteristic, complemented
4#14521367C211yesmaximal, characteristic, complemented
5#14521353C211yesmaximal, characteristic, complemented
6#14521283C211yesmaximal, characteristic, complemented
7#14521276C211yesmaximal, characteristic, complemented
8#14521264C211yesmaximal, characteristic, complemented
9#14521192C211yesmaximal, characteristic, complemented
10#14521187C211yesmaximal, characteristic, complemented
11C4×#92480C211yesmaximal, characteristic, complemented
12D4×C31⋊C8C211yesmaximal, characteristic, complemented
13#14521043C211yesmaximal, characteristic, complemented
14C2×#11758398C211yesmaximal, characteristic, complemented
15C31×#553C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C22×Dic62C2211yescharacteristic
18C31×#126—21no—
19C31×#125C2211yescharacteristic
20C31×#123—21no—
21C31×#121—21no—
22C31×#121C2211yescharacteristic
23C31×#121C2211yescharacteristic
24C4×D4×C31—21no—
25C4×D4×C31C2211yescharacteristic
26C4×D4×C31C2211yescharacteristic
27C31×#117C2211yescharacteristic
28C2×C4⋊C4×C31C2211yescharacteristic
29#11758516—21no—
30C2×#92480C2211yescharacteristic
31C2×#92480C2211yescharacteristic
32C2×#92479C2211yescharacteristic
33#11758507C2211yescharacteristic
34#11758501C2211yescharacteristic
35C22×C31⋊C8C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37#11758473—21no—
38#11758462—21no—
39(C22⋊D31)⋊C4—21no—
40#11758452C2211yescharacteristic
41C4×Dic62C2211yescharacteristic
42#11758420C2211yescharacteristic
43C22⋊(C31⋊C8)C2211yescharacteristic
44C22⋊(C31⋊C8)C2211yescharacteristic
45#11758398C2211yescharacteristic
46#11758398C2211yescharacteristic
47#11758398C2211yescharacteristic
48#11758398C2211yescharacteristic
49#11758398C2211yescharacteristic
50#11758398C2211yescharacteristic
51#11758397C2211yescharacteristic
52#11758397C2211yescharacteristic
53#11758397C2211yescharacteristic
54#11758396C2211yescharacteristic
55#11758396C2211yescharacteristic
56#11758395C2211yescharacteristic
57C4⋊(C31⋊C8)C2211yescharacteristic
58C4×C31⋊C8C2211yescharacteristic
59C2×D4×C31—21no—
60C2×D4×C31—21no—
61C2×D4×C31D411yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31C2311yescharacteristic
66C22×C4×C31C2311yescharacteristic
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C22×Dic31—21no—
70#92495—22no—
71#92495—41no—
72C2×D124C2311yescharacteristic
73C2×C4×D31—21no—
74C2×Dic62—22no—
75C2×Dic62—21no—
76C2×Dic62—41no—
77C2×Dic62C2311yescharacteristic
78C2×Dic62C2311yescharacteristic
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31D411yescharacteristic
83C4⋊C4×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C4⋊C4×C31C2311yescharacteristic
86C22⋊C4×C31—21no—
87C22⋊C4×C31—21no—
88C22⋊C4×C31—21no—
89C22⋊C4×C31—21no—
90C22⋊C4×C31D411yescharacteristic
91C22⋊C4×C31D411yescharacteristic
92C42×C31—21no—
93C42×C31C2311yescharacteristic
94C22⋊Dic31—21no—
95#92480—22no—
96#92480—41no—
97#92479—41no—
98(C2×D31)⋊C4—21no—
99(C2×D31)⋊C4—21no—
100(C2×D31)⋊C4—21no—
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104C4×Dic31—21no—
105C2×C31⋊C8—41no—
106C2×C31⋊C8—41no—
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C2×C31⋊C8C2311yescharacteristic
111C23×C31—21no—
112C23×C31C2×D411yescharacteristic
113C23×C31C2×D411yescharacteristic
114C22×D31—21no—
115D4×C31—22no—
116D4×C31—21no—
117D4×C31—21no—
118D4×C31—42no—
119D4×C31—81no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—81no—
128C2×C4×C31—81no—
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2411yescharacteristic
134C22⋊D31—42no—
135C22⋊D31—42no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31—21no—
139C2×Dic31—42no—
140D124—21no—
141D124—21no—
142C4×D31—42no—
143Dic62—21no—
144Dic62—21no—
145Dic62—21no—
146Dic62—42no—
147Dic62#4112yes—
148C31⋊C8—21no—
149C31⋊C8—21no—
150C31⋊C8—21no—
151C31⋊C8—21no—
152#3301—311nomaximal, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31#13512yes—
162C22×C31C22×D411yescharacteristic
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31C2×#4111yescharacteristic
174C4×C31H(5,𝔽2)11yescharacteristic
175Dic31—22no—
176Dic31—41no—
177Dic31—41no—
178#553D3111yescharacteristic, complemented
179#548—311no—
180#488—311no—
181#485—311no—
182#481—311no—
183#479—311no—
184#466—311no—
185#465—311no—
186#463—311no—
187#449—311no—
188#448—311no—
189C4×SD16—311no—
190C8×D4—311no—
191#416—311no—
192C2×#103—311no—
193C2×C31—22no—
194C2×C31—41no—
195C2×C31—81no—
196C2×C31#54611yescharacteristic
197C2×C31C2×#13511yescharacteristic
198C2×C31#57711yescharacteristic
199D31—81no—
200C2×#41—311no—
201C22×Q8—311no—
202C2×SD16—311no—
203C2×SD16—311no—
204C2×D8—311no—
205C22×C8—311no—
206C22×C8—311no—
207#126—21no—
208#125C2×D3111yescharacteristic
209#124—311no—
210#124—621no—
211#123—21no—
212C22⋊Q8—311no—
213C22⋊Q8—621no—
214#121—21no—
215#121—311no—
216#121—621no—
217#121C2×D3111yescharacteristic
218#121C2×D3111yescharacteristic
219C4×Q8—311no—
220C4×D4—21no—
221C4×D4—621no—
222C4×D4C2×D3111yescharacteristic
223C4×D4C2×D3111yescharacteristic
224#117C2×D3111yescharacteristic
225C2×C4⋊C4C2×D3111yescharacteristic
226#107—311no—
227#106—311no—
228#106—311no—
229C4⋊C8—311no—
230#103—311no—
231#103—311no—
232#103—311no—
233#103—311no—
234#103—311no—
235#103—311no—
236#102—311no—
237#102—311no—
238#102—311no—
239#102—311no—
240C22⋊C8—311no—
241C22⋊C8—311no—
242C4×C8—311no—
243C31#330111yescharacteristic, complemented, Sylow
244#41—622no—
245#41—1241no—
246C2×Q8—311no—
247C2×Q8—311no—
248C2×Q8—622no—
249C2×Q8—621no—
250C2×Q8—1241no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—311no—
254C2×D4—621no—
255C2×D4—621no—
256C2×D4C22⋊D3111yescharacteristic
257C2×D4C22×D3111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—621no—
261C22×C4—621no—
262C22×C4C22×D3111yescharacteristic
263C22×C4C22×D3111yescharacteristic
264SD16—622no—
265SD16—1241no—
266D8—1241no—
267C2×C8—311no—
268C2×C8—311no—
269C2×C8—311no—
270C2×C8—311no—
271C2×C8—1241no—
272C2×C8—1241no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—311no—
277C4⋊C4—621no—
278C4⋊C4—621no—
279C4⋊C4C22⋊D3111yescharacteristic
280C4⋊C4C22×D3111yescharacteristic
281C4⋊C4C22×D3111yescharacteristic
282C4⋊C4C22×D3111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C22⋊C4—621no—
290C22⋊C4—621no—
291C22⋊C4C22⋊D3111yescharacteristic
292C22⋊C4C22⋊D3111yescharacteristic
293C42—21no—
294C42—621no—
295C42C22×D3111yescharacteristic
296C23—21no—
297C23—621no—
298C23C2×C22⋊D3111yescharacteristic
299C23C2×C22⋊D3111yescharacteristic
300Q8—312no—
301Q8—621no—
302Q8—621no—
303Q8—621no—
304Q8—1242no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—621no—
311D4—621no—
312D4—1242no—
313D4—1242no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—621no—
326C2×C4—1242no—
327C2×C4—1242no—
328C2×C4C2×C22⋊D3111yescharacteristic
329C2×C4C2×C22⋊D3111yescharacteristic
330C2×C4C2×C22⋊D3111yescharacteristic
331C2×C4C2×C22⋊D3111yescharacteristic
332C2×C4C23×D3111yescharacteristic
333C8—621no—
334C8—621no—
335C8—621no—
336C8—621no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—1241no—
346C22—1241no—
347C22—1241no—
348C22#1175852712yes—
349C22C22×C22⋊D3111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—622no—
358C4—1241no—
359C4—1241no—
360C4#41×D3111yescharacteristic
361C4#1175856611yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—2481no—
366C2#1452168711yescharacteristic
367C2C2×#1175852711yescharacteristic
368C2#1452186011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#553C2
socleC22×C31C22×D4
hypercenter#553D31
hypocenterC31#3301
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #14521708C2semidirect
2. #14521406C2semidirect
3. #14521390C2semidirect
4. #14521367C2semidirect
5. #14521353C2semidirect
6. #14521283C2semidirect
7. #14521276C2semidirect
8. #14521264C2semidirect
9. #14521192C2semidirect
10. #14521187C2semidirect
11. C4×#92480C2semidirect
12. D4×C31⋊C8C2semidirect
13. #14521043C2semidirect
14. C2×#11758398C2semidirect
15. C31×#553C2semidirect
178. #553D31semidirect
243. C31#3301semidirect

Maximal subgroups

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

subgroupindex
#145217082
#145214062
#145213902
#145213672
#145213532
#145212832
#145212762
#145212642
#145211922
#145211872
C4×#924802
D4×C31⋊C82
#145210432
C2×#117583982
C31×#5532
#330131