Finite Group Databasebeta

← C31×#10016 #21466775

Subgroups — C31×#10016 #21466775

Order 7936.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3104C211yesmaximal, characteristic, complemented
2C31×#3101C211yesmaximal, characteristic
3C31×#2398C211yesmaximal, characteristic
4C31×#1995C211yesmaximal, characteristic, complemented
5C31×#1487C211yesmaximal, characteristic, complemented
6C31×#1486C211yesmaximal, characteristic
7C31×#1274C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C4×Q8×C31C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#484—21no—
12C31×#483—21no—
13C31×#477—21no—
14C31×#476—21no—
15C31×#462—21no—
16C31×#461—21no—
17C31×#451—21no—
18C31×#450—21no—
19C4×Q16×C31—22no—
20C4×SD16×C31—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C31×#400—21no—
25C31×#397—21no—
26C31×#394—21no—
27C31×#393—21no—
28C4×C22⋊C4×C31—22no—
29C31×#376C2211yescharacteristic
30C31×#339—22no—
31C31×Q8⋊C8—22no—
32C31×D4⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C2×Q16×C31—22no—
37C2×SD16×C31—22no—
38C2×M16×C31—21no—
39C31×C4⋊Q8D411yescharacteristic
40C31×C4⋊Q8D411yescharacteristic
41C31×#125—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C4×Q8×C31—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C31×#117—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C22⋊C4×C31—22no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—21no—
61C2×C42×C31—21no—
62C2×C42×C31D412yes—
63C2×C42×C31C2311yescharacteristic
64C31×#107—22no—
65C31×#106—22no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8—21no—
70C31×#103—22no—
71C31×#103—22no—
72C31×#103—22no—
73C31×#102—22no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C4×C8×C31D412yes—
77C4×C8×C31D412yes—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—21no—
81C31×#95—21no—
82C24×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—42no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—42no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31—42no—
98C22×C4×C31—42no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102Q16×C31—44no—
103SD16×C31—44no—
104M16×C31—42no—
105C2×C8×C31—22no—
106C2×C8×C31—22no—
107C2×C8×C31—41no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—44no—
121C22⋊C4×C31—44no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C42×C31—22no—
126C42×C31—22no—
127C42×C31—41no—
128C42×C31—41no—
129C42×C31—41no—
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132C42×C31C2×D411yescharacteristic
133C42×C31#4111yescharacteristic
134#10016C3111yesmaximal, characteristic, direct product factor, Sylow
135C23×C31—22no—
136C23×C31—42no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—42no—
140C23×C31#12011yescharacteristic
141Q8×C31—22no—
142Q8×C31—22no—
143Q8×C31—41no—
144Q8×C31—41no—
145Q8×C31—42no—
146Q8×C31—42no—
147D4×C31—22no—
148D4×C31—22no—
149D4×C31—41no—
150D4×C31—41no—
151C2×C4×C31—22no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—82no—
166C2×C4×C31—81no—
167C2×C4×C31—81no—
168C2×C4×C31#12011yescharacteristic
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12111yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12711yescharacteristic
174C2×C4×C31#13512yes—
175C2×C4×C31#13712yes—
176C8×C31—42no—
177C8×C31—42no—
178C8×C31—81no—
179#3104C2×C3111yescharacteristic, complemented
180#3101C2×C3111yescharacteristic
181#2398C2×C3111yescharacteristic
182#1995C2×C3111yescharacteristic, complemented
183#1487C2×C3111yescharacteristic, complemented
184#1486C2×C3111yescharacteristic
185#1274C2×C3111yescharacteristic, complemented
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—44no—
189C22×C31—42no—
190C22×C31—82no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31#39211yescharacteristic
194C22×C31#45212yes—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—42no—
200C4×C31—82no—
201C4×C31—81no—
202C4×C31—81no—
203C4×C31—81no—
204C4×C31—81no—
205C4×C31#46412yes—
206C4×C31#49511yescharacteristic
207C4×C31#49711yescharacteristic
208#533C22×C3111yescharacteristic
209C2×C4×Q8C22×C3111yescharacteristic
210C2×C4×D4C22×C3111yescharacteristic
211#484—21no—
212#483—21no—
213#477—21no—
214#476—21no—
215#462—21no—
216#461—21no—
217#451—21no—
218#450—21no—
219C4×Q16—22no—
220C4×SD16—22no—
221#431C22×C3111yescharacteristic
222#431C22×C3111yescharacteristic
223#423C22×C3111yescharacteristic
224#400—21no—
225#397—21no—
226#394—21no—
227#393—21no—
228C4×C22⋊C4—22no—
229#376C22×C3111yescharacteristic
230#339—22no—
231Q8⋊C8—22no—
232D4⋊C8—22no—
233C2×C31—22no—
234C2×C31—82no—
235C2×C31#201011yescharacteristic
236C2×C31#201011yescharacteristic
237C2×C31#201511yescharacteristic
238C22×Q8D4×C3111yescharacteristic
239C22×D4D4×C3111yescharacteristic
240C23×C4—21no—
241C2×Q16—22no—
242C2×SD16—22no—
243C2×M16—21no—
244C4⋊Q8D4×C3111yescharacteristic
245C4⋊Q8D4×C3111yescharacteristic
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259#117—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4D4×C3111yescharacteristic
262C2×C4⋊C4D4×C3111yescharacteristic
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C42—21no—
267C2×C42D4×C3112yes—
268C2×C42C23×C3111yescharacteristic
269#107—22no—
270#106—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275#103—22no—
276#103—22no—
277#103—22no—
278#102—22no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C4×C8D4×C3112yes—
282C4×C8D4×C3112yes—
283#95—22no—
284#95—22no—
285#95—21no—
286#95—21no—
287C31#1001611yescharacteristic, direct product factor, Sylow
288C24—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4C2×D4×C3111yescharacteristic
306C22×C4C2×D4×C3111yescharacteristic
307C22×C4C2×D4×C3111yescharacteristic
308Q16—44no—
309SD16—44no—
310M16—42no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42—22no—
332C42—22no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42C2×D4×C3111yescharacteristic
337C42C2×D4×C3111yescharacteristic
338C42C2×D4×C3111yescharacteristic
339C42#41×C3111yescharacteristic
340C23—22no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23C31×#12011yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350Q8—42no—
351Q8—42no—
352D4—22no—
353D4—22no—
354D4—41no—
355D4—41no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4C31×#12011yescharacteristic
374C2×C4C31×#12011yescharacteristic
375C2×C4C31×#12111yescharacteristic
376C2×C4C31×#12111yescharacteristic
377C2×C4C31×#12111yescharacteristic
378C2×C4C31×#12711yescharacteristic
379C2×C4C31×#13512yes—
380C2×C4C31×#13712yes—
381C8—42no—
382C8—42no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—42no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22C31×#39211yescharacteristic
392C22C31×#45212yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4C31×#46412yes—
404C4C31×#49511yescharacteristic
405C4C31×#49711yescharacteristic
406C2—22no—
407C2—82no—
408C2C31×#201011yescharacteristic
409C2C31×#201011yescharacteristic
410C2C31×#201511yescharacteristic
subgroupG/H
centerC2×C4×C31#127
commutator, Frattini, cosocleC2×C42C23×C31
socleC22×C31#392
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3104C2semidirect
4. C31×#1995C2semidirect
5. C31×#1487C2semidirect
7. C31×#1274C2semidirect
134. #10016C31direct (N × H)
179. #3104C2×C31semidirect
182. #1995C2×C31semidirect
183. #1487C2×C31semidirect
185. #1274C2×C31semidirect
287. C31#10016direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31042
C31×#31012
C31×#23982
C31×#19952
C31×#14872
C31×#14862
C31×#12742
#1001631