Finite Group Databasebeta

← C31×#12102 #21468861

Subgroups — C31×#12102 #21468861

Order 7936.

528 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C31×#3186C211yesmaximal, characteristic, complemented
2C31×#2184C212yesmaximal, complemented
3C31×#1999C211yesmaximal, characteristic, complemented
4C31×#1347C212yesmaximal, complemented
5C31×#1314C211yesmaximal, characteristic, complemented
6C22×D8×C31C2211yescharacteristic, complemented
7C2×C31×#121—21no—
8C2×C31×#121—21no—
9C2×C31×D16—22no—
10C31×#480—22no—
11C31×#466—21no—
12C31×#466C2212yes—
13C31×#447—22no—
14C2×C31×#102—21no—
15C2×C31×C22⋊C8C2211yescharacteristic
16C31×#357—22no—
17C31×C22⋊C16C2212yescomplemented
18C31×#340C2211yescharacteristic, complemented
19C22×D4×C31—21no—
20C22×D4×C31—21no—
21C22×D4×C31—41no—
22C22×D4×C31—41no—
23C23×C4×C31D411yescharacteristic
24C2×D8×C31—21no—
25C2×D8×C31—22no—
26C2×D8×C31—42no—
27C2×D8×C31D412yescomplemented
28C22×C8×C31D411yescharacteristic
29C22×C8×C31C2311yescharacteristic
30C31×#121—22no—
31C31×#121—22no—
32C31×#121—42no—
33C31×#121—41no—
34C31×#121—42no—
35C31×#121—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C22⋊C4×C31—41no—
39C2×C22⋊C4×C31—41no—
40C31×D16—44no—
41C2×C16×C31—22no—
42C31×#107—21no—
43C31×#107D412yescomplemented
44C31×#102—22no—
45C31×#102—22no—
46C31×#102—42no—
47C31×C22⋊C8—22no—
48C24×C31—41no—
49C24×C31—41no—
50C24×C31D811yescharacteristic
51C2×D4×C31—22no—
52C2×D4×C31—22no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31—42no—
58C2×D4×C31—44no—
59C2×D4×C31—42no—
60C2×D4×C31—41no—
61C2×D4×C31—41no—
62C2×D4×C31—81no—
63C2×D4×C31—81no—
64C2×D4×C31—82no—
65C2×D4×C31—82no—
66C2×D4×C31—82no—
67C2×D4×C31—82no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—41no—
71C22×C4×C31—41no—
72C22×C4×C31—41no—
73C22×C4×C31—41no—
74C22×C4×C31D811yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76D8×C31—24no—
77D8×C31—42no—
78D8×C31—42no—
79D8×C31—81no—
80C2×C8×C31—21no—
81C2×C8×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31C2×D412yes—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—44no—
90C22⋊C4×C31—42no—
91C16×C31—42no—
92#12102C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—21no—
94C23×C31—21no—
95C23×C31—41no—
96C23×C31—41no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—81no—
100C23×C31—81no—
101C23×C31—82no—
102C23×C31—81no—
103C23×C31—82no—
104C23×C31—82no—
105C23×C31—81no—
106C23×C31—81no—
107C23×C31—81no—
108C23×C31D1612yescomplemented
109C23×C31C2×D811yescharacteristic
110D4×C31—42no—
111D4×C31—42no—
112D4×C31—42no—
113D4×C31—42no—
114D4×C31—81no—
115D4×C31—82no—
116D4×C31—84no—
117D4×C31—88no—
118D4×C31—82no—
119D4×C31—82no—
120D4×C31—81no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—81no—
130C2×C4×C31—81no—
131C2×C4×C31—82no—
132C2×C4×C31—81no—
133C2×C4×C31#12011yescharacteristic
134C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
135C8×C31—22no—
136C8×C31—41no—
137#3186C2×C3111yescharacteristic, complemented
138#2184C2×C3112yescomplemented
139#1999C2×C3111yescharacteristic, complemented
140#1347C2×C3112yescomplemented
141#1314C2×C3111yescharacteristic, complemented
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—22no—
145C22×C31—42no—
146C22×C31—41no—
147C22×C31—41no—
148C22×C31—81no—
149C22×C31—82no—
150C22×C31—82no—
151C22×C31—82no—
152C22×C31—82no—
153C22×C31—82no—
154C22×C31—82no—
155C22×C31—161no—
156C22×C31—161no—
157C22×C31—164no—
158C22×C31—161no—
159C22×C31—161no—
160C22×C31#44711yescharacteristic
161C22×C31C2×D1611yescharacteristic
162C22×C31#50911yescharacteristic
163C4×C31—21no—
164C4×C31—41no—
165C4×C31—82no—
166C4×C31—82no—
167C4×C31#45311yescharacteristic
168C4×C31B(4,𝔽2)11yescharacteristic
169C22×D8C22×C3111yescharacteristic, complemented
170C2×#121—21no—
171C2×#121—21no—
172C2×D16—22no—
173#480—22no—
174#466—21no—
175#466C22×C3112yes—
176#447—22no—
177C2×#102—21no—
178C2×C22⋊C8C22×C3111yescharacteristic
179#357—22no—
180C22⋊C16C22×C3112yescomplemented
181#340C22×C3111yescharacteristic, complemented
182C2×C31—21no—
183C2×C31—21no—
184C2×C31—42no—
185C2×C31—162no—
186C2×C31—162no—
187C2×C31#159511yescharacteristic
188C2×C31#218411yescharacteristic
189C2×C31#219011yescharacteristic
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—41no—
193C22×D4—41no—
194C23×C4D4×C3111yescharacteristic
195C2×D8—21no—
196C2×D8—22no—
197C2×D8—42no—
198C2×D8D4×C3112yescomplemented
199C22×C8D4×C3111yescharacteristic
200C22×C8C23×C3111yescharacteristic
201#121—22no—
202#121—22no—
203#121—42no—
204#121—41no—
205#121—42no—
206#121—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211D16—44no—
212C2×C16—22no—
213#107—21no—
214#107D4×C3112yescomplemented
215#102—22no—
216#102—22no—
217#102—42no—
218C22⋊C8—22no—
219C31#1210211yescharacteristic, direct product factor, Sylow
220C24—41no—
221C24—41no—
222C24D8×C3111yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—81no—
235C2×D4—81no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—82no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4D8×C3111yescharacteristic
247C22×C4C2×D4×C3111yescharacteristic
248D8—24no—
249D8—42no—
250D8—42no—
251D8—81no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8C2×D4×C3112yes—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C16—42no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—82no—
273C23—81no—
274C23—82no—
275C23—82no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23C31×D1612yescomplemented
280C23C2×D8×C3111yescharacteristic
281D4—42no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—81no—
286D4—82no—
287D4—84no—
288D4—88no—
289D4—82no—
290D4—82no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4C31×#12011yescharacteristic
305C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
306C8—22no—
307C8—41no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—161no—
322C22—161no—
323C22—164no—
324C22—161no—
325C22—161no—
326C22C31×#44711yescharacteristic
327C22C2×C31×D1611yescharacteristic
328C22C31×#50911yescharacteristic
329C4—21no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4C31×#45311yescharacteristic
334C4C31×B(4,𝔽2)11yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2—162no—
339C2—162no—
340C2C31×#159511yescharacteristic
341C2C31×#218411yescharacteristic
342C2C31×#219011yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×C22⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3186C2semidirect
2. C31×#2184C2semidirect
3. C31×#1999C2semidirect
4. C31×#1347C2semidirect
5. C31×#1314C2semidirect
6. C22×D8×C31C22semidirect
17. C31×C22⋊C16C22semidirect
18. C31×#340C22semidirect
27. C2×D8×C31D4semidirect
43. C31×#107D4semidirect
92. #12102C31direct (N × H)
108. C23×C31D16semidirect
137. #3186C2×C31semidirect
138. #2184C2×C31semidirect
139. #1999C2×C31semidirect
140. #1347C2×C31semidirect
141. #1314C2×C31semidirect
169. C22×D8C22×C31semidirect
180. C22⋊C16C22×C31semidirect
181. #340C22×C31semidirect
198. C2×D8D4×C31semidirect
214. #107D4×C31semidirect
219. C31#12102direct (N × H)
279. C23C31×D16semidirect

Maximal subgroups

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

subgroupindex
C31×#31862
C31×#21842
C31×#19992
C31×#13472
C31×#13142
#1210231