Finite Group Databasebeta

← C31×#8522 #21465281

Subgroups — C31×#8522 #21465281

Order 7936.

1102 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 150 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×#535C211yesmaximal, characteristic
2C31×#1555C212yesmaximal, complemented
3C31×#1533C212yesmaximal, complemented
4C31×#1464C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic
6C23×D4×C31C411yescharacteristic
7C31×#535—24no—
8C31×#535—24no—
9C2×C31×#127C411yescharacteristic
10C2×C31×#127C2211yescharacteristic
11C2×C31×#124C411yescharacteristic
12C2×C31×#124C2211yescharacteristic
13C2×C31×#121—22no—
14C2×C31×#120—22no—
15C2×C31×#117C2211yescharacteristic
16C31×#432C2214yescomplemented
17C31×#331—24no—
18C31×#329—24no—
19C31×#324—22no—
20C31×#323—22no—
21C25×C31—21no—
22C2×#41×C31—22no—
23C2×#41×C31—22no—
24C2×#41×C31—42no—
25C2×#41×C31—42no—
26C22×Q8×C31C2×C411yescharacteristic
27C22×D4×C31—21no—
28C22×D4×C31—22no—
29C22×D4×C31—22no—
30C22×D4×C31—21no—
31C22×D4×C31—22no—
32C22×D4×C31—22no—
33C22×D4×C31—42no—
34C22×D4×C31—42no—
35C22×D4×C31C2×C411yescharacteristic
36C22×D4×C31C2×C411yescharacteristic
37C22×D4×C31C2×C411yescharacteristic
38C23×C4×C31—21no—
39C23×C4×C31C2×C411yescharacteristic
40C31×#127—22no—
41C31×#127—22no—
42C31×#127—22no—
43C31×#127D414yes—
44C31×#124—22no—
45C31×#124—22no—
46C31×#124—22no—
47C31×#124D414yes—
48C31×#121—216no—
49C31×#120—216no—
50C31×#117—22no—
51C31×#117—22no—
52C2×C4⋊C4×C31D412yes—
53C2×C22⋊C4×C31—22no—
54C2×C22⋊C4×C31—22no—
55C2×C22⋊C4×C31D412yes—
56C2×C42×C31C2×C411yescharacteristic
57C2×C42×C31C2311yescharacteristic
58C31×C4⋊C8—24no—
59C31×C22⋊C8—24no—
60C31×#97—24no—
61C24×C31—24no—
62C24×C31—21no—
63C24×C31—21no—
64C24×C31—21no—
65C24×C31—42no—
66C24×C31—42no—
67C24×C31—84no—
68C24×C31C22⋊C411yescharacteristic
69#41×C31—48no—
70#41×C31—48no—
71C2×Q8×C31—21no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×Q8×C31C22⋊C412yes—
75C2×D4×C31—24no—
76C2×D4×C31—24no—
77C2×D4×C31—28no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—28no—
82C2×D4×C31—48no—
83C2×D4×C31—48no—
84C2×D4×C31—42no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—42no—
92C2×D4×C31—42no—
93C2×D4×C31—42no—
94C2×D4×C31—84no—
95C2×D4×C31—84no—
96C2×D4×C31C22⋊C412yes—
97C2×D4×C31C22⋊C412yes—
98C2×D4×C31C22⋊C412yes—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—24no—
104C22×C4×C31—21no—
105C22×C4×C31—42no—
106C22×C4×C31—42no—
107C22×C4×C31C22⋊C411yescharacteristic
108C22×C4×C31C22⋊C412yes—
109C22×C4×C31C22×C411yescharacteristic
110C22×C4×C31C2×D412yes—
111C2×C8×C31—44no—
112C4⋊C4×C31—24no—
113C22⋊C4×C31—28no—
114C22⋊C4×C31—28no—
115C22⋊C4×C31—24no—
116C42×C31—21no—
117C42×C31—21no—
118C42×C31C2×D412yes—
119C42×C31C2×D412yes—
120#8522C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—21no—
122C23×C31—21no—
123C23×C31—22no—
124C23×C31—22no—
125C23×C31—22no—
126C23×C31—42no—
127C23×C31—48no—
128C23×C31—44no—
129C23×C31—44no—
130C23×C31—44no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—88no—
138C23×C31—88no—
139C23×C31—88no—
140C23×C31#10012yes—
141C23×C31C2×C22⋊C411yescharacteristic
142Q8×C31—42no—
143Q8×C31—42no—
144D4×C31—42no—
145D4×C31—42no—
146D4×C31—42no—
147D4×C31—48no—
148D4×C31—42no—
149D4×C31—42no—
150D4×C31—42no—
151D4×C31—48no—
152D4×C31—88no—
153D4×C31—88no—
154C2×C4×C31—24no—
155C2×C4×C31—24no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—22no—
161C2×C4×C31—22no—
162C2×C4×C31—42no—
163C2×C4×C31—48no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31#10012yes—
168C2×C4×C31C2×C22⋊C411yescharacteristic
169C2×C4×C31C2×C22⋊C411yescharacteristic
170C2×C4×C31#12014yes—
171C8×C31—84no—
172C2×#535C2×C3111yescharacteristic
173#1555C2×C3112yescomplemented
174#1533C2×C3112yescomplemented
175#1464C2×C3112yescomplemented
176C22×C31—22no—
177C22×C31—22no—
178C22×C31—42no—
179C22×C31—42no—
180C22×C31—42no—
181C22×C31—44no—
182C22×C31—44no—
183C22×C31—44no—
184C22×C31—42no—
185C22×C31—42no—
186C22×C31—42no—
187C22×C31—82no—
188C22×C31—82no—
189C22×C31—82no—
190C22×C31—82no—
191C22×C31—82no—
192C22×C31—82no—
193C22×C31—816no—
194C22×C31#37911yescharacteristic
195C22×C31C2×#10011yescharacteristic
196C22×C31C2×#10011yescharacteristic
197C4×C31—24no—
198C4×C31—22no—
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—42no—
202C4×C31—42no—
203C22×#41C4×C3111yescharacteristic
204C23×D4C4×C3111yescharacteristic
205#535—24no—
206#535—24no—
207C2×#127C4×C3111yescharacteristic
208C2×#127C22×C3111yescharacteristic
209C2×#124C4×C3111yescharacteristic
210C2×#124C22×C3111yescharacteristic
211C2×#121—22no—
212C2×#120—22no—
213C2×#117C22×C3111yescharacteristic
214#432C22×C3114yescomplemented
215#331—24no—
216#329—24no—
217#324—22no—
218#323—22no—
219C2×C31—21no—
220C2×C31—21no—
221C2×C31—42no—
222C2×C31—82no—
223C2×C31—82no—
224C2×C31—82no—
225C2×C31#178311yescharacteristic
226C2×C31#179512yes—
227C25—21no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C22×Q8C2×C4×C3111yescharacteristic
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4C2×C4×C3111yescharacteristic
242C22×D4C2×C4×C3111yescharacteristic
243C22×D4C2×C4×C3111yescharacteristic
244C23×C4—21no—
245C23×C4C2×C4×C3111yescharacteristic
246#127—22no—
247#127—22no—
248#127—22no—
249#127D4×C3114yes—
250#124—22no—
251#124—22no—
252#124—22no—
253#124D4×C3114yes—
254#121—216no—
255#120—216no—
256#117—22no—
257#117—22no—
258C2×C4⋊C4D4×C3112yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4D4×C3112yes—
262C2×C42C2×C4×C3111yescharacteristic
263C2×C42C23×C3111yescharacteristic
264C4⋊C8—24no—
265C22⋊C8—24no—
266#97—24no—
267C31#852211yescharacteristic, direct product factor, Sylow
268C24—24no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—42no—
273C24—42no—
274C24—84no—
275C24C22⋊C4×C3111yescharacteristic
276#41—48no—
277#41—48no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C3112yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—28no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4C22⋊C4×C3112yes—
304C2×D4C22⋊C4×C3112yes—
305C2×D4C22⋊C4×C3112yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C22⋊C4×C3111yescharacteristic
315C22×C4C22⋊C4×C3112yes—
316C22×C4C22×C4×C3111yescharacteristic
317C22×C4C2×D4×C3112yes—
318C2×C8—44no—
319C4⋊C4—24no—
320C22⋊C4—28no—
321C22⋊C4—28no—
322C22⋊C4—24no—
323C42—21no—
324C42—21no—
325C42C2×D4×C3112yes—
326C42C2×D4×C3112yes—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—42no—
333C23—48no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—88no—
344C23—88no—
345C23—88no—
346C23C31×#10012yes—
347C23C2×C22⋊C4×C3111yescharacteristic
348Q8—42no—
349Q8—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—48no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—48no—
358D4—88no—
359D4—88no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4C31×#10012yes—
374C2×C4C2×C22⋊C4×C3111yescharacteristic
375C2×C4C2×C22⋊C4×C3111yescharacteristic
376C2×C4C31×#12014yes—
377C8—84no—
378C22—22no—
379C22—22no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—816no—
396C22C31×#37911yescharacteristic
397C22C2×C31×#10011yescharacteristic
398C22C2×C31×#10011yescharacteristic
399C4—24no—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C2—21no—
406C2—21no—
407C2—42no—
408C2—82no—
409C2—82no—
410C2—82no—
411C2C31×#178311yescharacteristic
412C2C31×#179512yes—
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1555C2semidirect
3. C31×#1533C2semidirect
4. C31×#1464C2semidirect
16. C31×#432C22semidirect
120. #8522C31direct (N × H)
173. #1555C2×C31semidirect
174. #1533C2×C31semidirect
175. #1464C2×C31semidirect
214. #432C22×C31semidirect
267. C31#8522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5352
C31×#15552
C31×#15332
C31×#14642
#852231