Finite Group Databasebeta

← C31×#9964 #21466723

Subgroups — C31×#9964 #21466723

Order 7936.

944 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×C31×#448C212yesmaximal, complemented
2C31×#2656C211yesmaximal, characteristic
3C2×C31×#327C212yesmaximal, complemented
4C31×#1446C211yesmaximal, characteristic, complemented
5C31×#1280C211yesmaximal, characteristic, complemented
6C23×Q8×C31C2211yescharacteristic
7C22×SD16×C31—22no—
8C2×C31×#121—22no—
9C2×C31×#121C2212yes—
10C31×#448—24no—
11C31×#448—24no—
12C2×C31×#103—22no—
13C2×C31×C22⋊C8C2212yes—
14C31×#399—22no—
15C31×#392—22no—
16C2×C31×#95—22no—
17C2×C31×#95C2211yescharacteristic
18C2×C31×#95C2211yescharacteristic
19C31×#327—24no—
20C22×Q8×C31—21no—
21C22×Q8×C31—22no—
22C22×Q8×C31—22no—
23C22×Q8×C31—44no—
24C22×Q8×C31D412yescomplemented
25C22×D4×C31—22no—
26C22×D4×C31—22no—
27C22×D4×C31D412yes—
28C23×C4×C31—21no—
29C23×C4×C31—21no—
30C23×C4×C31D412yes—
31C23×C4×C31C2311yescharacteristic
32C2×SD16×C31—28no—
33C2×SD16×C31—48no—
34C22×C8×C31—22no—
35C31×#121—22no—
36C31×#121—22no—
37C31×#121—22no—
38C31×#121—22no—
39C31×#121—44no—
40C31×#121—44no—
41C2×C4⋊C4×C31—22no—
42C2×C4⋊C4×C31D412yes—
43C2×C22⋊C4×C31—24no—
44C2×C22⋊C4×C31—22no—
45C31×#103—28no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8D414yescomplemented
48C31×#95—22no—
49C31×#95—22no—
50C31×#95—44no—
51C24×C31—22no—
52C24×C31C2×D411yescharacteristic
53C2×Q8×C31—28no—
54C2×Q8×C31—24no—
55C2×Q8×C31—44no—
56C2×Q8×C31—44no—
57C2×Q8×C31—44no—
58C2×Q8×C31—48no—
59C2×Q8×C31—42no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—44no—
63C2×D4×C31—44no—
64C2×D4×C31—42no—
65C2×D4×C31—42no—
66C2×D4×C31—44no—
67C2×D4×C31—44no—
68C2×D4×C31—82no—
69C2×D4×C31—82no—
70C22×C4×C31—24no—
71C22×C4×C31—22no—
72C22×C4×C31—22no—
73C22×C4×C31—22no—
74C22×C4×C31—22no—
75C22×C4×C31—22no—
76C22×C4×C31—42no—
77C22×C4×C31—42no—
78C22×C4×C31—41no—
79C22×C4×C31—41no—
80C22×C4×C31—42no—
81C22×C4×C31—42no—
82C22×C4×C31—42no—
83C22×C4×C31—41no—
84C22×C4×C31—41no—
85C22×C4×C31—42no—
86C22×C4×C31SD1614yescomplemented
87C22×C4×C31C2×D412yes—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D412yes—
90C22×C4×C31#4111yescharacteristic
91SD16×C31—416no—
92C2×C8×C31—24no—
93C2×C8×C31—44no—
94C4⋊C4×C31—22no—
95C4⋊C4×C31—22no—
96C4⋊C4×C31—44no—
97C22⋊C4×C31—48no—
98C22⋊C4×C31—42no—
99C22⋊C4×C31—42no—
100#9964C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—22no—
104C23×C31—42no—
105C23×C31—44no—
106C23×C31—42no—
107C23×C31—42no—
108C23×C31—44no—
109C23×C31#12011yescharacteristic
110C23×C31#12111yescharacteristic
111C23×C31#12711yescharacteristic
112C23×C31C2×SD1612yes—
113C23×C31AGL(1,ℤ/8)12yes—
114Q8×C31—28no—
115Q8×C31—44no—
116Q8×C31—48no—
117D4×C31—44no—
118D4×C31—82no—
119D4×C31—88no—
120D4×C31—84no—
121D4×C31—82no—
122D4×C31—82no—
123C2×C4×C31—24no—
124C2×C4×C31—24no—
125C2×C4×C31—24no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—41no—
129C2×C4×C31—42no—
130C2×C4×C31—44no—
131C2×C4×C31—42no—
132C2×C4×C31—42no—
133C2×C4×C31—44no—
134C2×C4×C31—42no—
135C2×C4×C31—44no—
136C2×C4×C31—44no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—44no—
140C2×C4×C31—82no—
141C2×C4×C31—82no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31#12012yes—
145C2×C4×C31#12112yes—
146C8×C31—44no—
147C2×#448C2×C3112yescomplemented
148#2656C2×C3111yescharacteristic
149C2×#327C2×C3112yescomplemented
150#1446C2×C3111yescharacteristic, complemented
151#1280C2×C3111yescharacteristic, complemented
152C22×C31—22no—
153C22×C31—22no—
154C22×C31—22no—
155C22×C31—22no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—84no—
161C22×C31—84no—
162C22×C31—82no—
163C22×C31—82no—
164C22×C31—82no—
165C22×C31#39211yescharacteristic
166C22×C31#45012yes—
167C22×C31#45511yescharacteristic
168C22×C31#45611yescharacteristic
169C22×C31B(4,𝔽2)12yes—
170C22×C31#46012yes—
171C22×C31#49211yescharacteristic
172C22×C31#49611yescharacteristic
173C4×C31—22no—
174C4×C31—22no—
175C4×C31—44no—
176C4×C31—44no—
177C4×C31—82no—
178C4×C31—82no—
179C23×Q8C22×C3111yescharacteristic
180C22×SD16—22no—
181C2×#121—22no—
182C2×#121C22×C3112yes—
183#448—24no—
184#448—24no—
185C2×#103—22no—
186C2×C22⋊C8C22×C3112yes—
187#399—22no—
188#392—22no—
189C2×#95—22no—
190C2×#95C22×C3111yescharacteristic
191C2×#95C22×C3111yescharacteristic
192#327—24no—
193C2×C31—21no—
194C2×C31—22no—
195C2×C31—21no—
196C2×C31—162no—
197C2×C31#160112yes—
198C2×C31#160412yes—
199C2×C31#200711yescharacteristic
200C2×C31#201411yescharacteristic
201C2×C31#202111yescharacteristic
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—44no—
206C22×Q8D4×C3112yescomplemented
207C22×D4—22no—
208C22×D4—22no—
209C22×D4D4×C3112yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4D4×C3112yes—
213C23×C4C23×C3111yescharacteristic
214C2×SD16—28no—
215C2×SD16—48no—
216C22×C8—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—22no—
221#121—44no—
222#121—44no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4D4×C3112yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227#103—28no—
228C22⋊C8—22no—
229C22⋊C8D4×C3114yescomplemented
230#95—22no—
231#95—22no—
232#95—44no—
233C31#996411yescharacteristic, direct product factor, Sylow
234C24—22no—
235C24C2×D4×C3111yescharacteristic
236C2×Q8—28no—
237C2×Q8—24no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—82no—
252C2×D4—82no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4SD16×C3114yescomplemented
270C22×C4C2×D4×C3112yes—
271C22×C4C2×D4×C3111yescharacteristic
272C22×C4C2×D4×C3112yes—
273C22×C4#41×C3111yescharacteristic
274SD16—416no—
275C2×C8—24no—
276C2×C8—44no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C22⋊C4—48no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—42no—
288C23—42no—
289C23—44no—
290C23—42no—
291C23C31×#12011yescharacteristic
292C23C31×#12111yescharacteristic
293C23C31×#12711yescharacteristic
294C23C2×SD16×C3112yes—
295C23C31×AGL(1,ℤ/8)12yes—
296Q8—28no—
297Q8—44no—
298Q8—48no—
299D4—44no—
300D4—82no—
301D4—88no—
302D4—84no—
303D4—82no—
304D4—82no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4C31×#12012yes—
327C2×C4C31×#12112yes—
328C8—44no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—84no—
338C22—84no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22C31×#39211yescharacteristic
343C22C31×#45012yes—
344C22C31×#45511yescharacteristic
345C22C31×#45611yescharacteristic
346C22C31×B(4,𝔽2)12yes—
347C22C31×#46012yes—
348C22C31×#49211yescharacteristic
349C22C31×#49611yescharacteristic
350C4—22no—
351C4—22no—
352C4—44no—
353C4—44no—
354C4—82no—
355C4—82no—
356C2—21no—
357C2—22no—
358C2—21no—
359C2—162no—
360C2C31×#160112yes—
361C2C31×#160412yes—
362C2C31×#200711yescharacteristic
363C2C31×#201411yescharacteristic
364C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#448C2semidirect
3. C2×C31×#327C2semidirect
4. C31×#1446C2semidirect
5. C31×#1280C2semidirect
24. C22×Q8×C31D4semidirect
47. C31×C22⋊C8D4semidirect
86. C22×C4×C31SD16semidirect
100. #9964C31direct (N × H)
147. C2×#448C2×C31semidirect
149. C2×#327C2×C31semidirect
150. #1446C2×C31semidirect
151. #1280C2×C31semidirect
206. C22×Q8D4×C31semidirect
229. C22⋊C8D4×C31semidirect
233. C31#9964direct (N × H)
269. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4482
C31×#26562
C2×C31×#3272
C31×#14462
C31×#12802
#996431