Finite Group Databasebeta

← #16857267

Subgroups — #16857267

Order 3968.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758519C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3#14521704C211yesmaximal, characteristic, complemented
4#14521444C211yesmaximal, characteristic, complemented
5#14521293C211yesmaximal, characteristic, complemented
6#14521288C211yesmaximal, characteristic, complemented
7#14521261C211yesmaximal, characteristic, complemented
8#14521259C211yesmaximal, characteristic, complemented
9#14521211C211yesmaximal, characteristic, complemented
10#14521209C211yesmaximal, characteristic, complemented
11#14521205C211yesmaximal, characteristic, complemented
12#14521192C211yesmaximal, characteristic, complemented
13#14521147C211yesmaximal, characteristic, complemented
14#14521121C211yesmaximal, characteristic, complemented
15C31×#548C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic
18C2×#92495C2211yescharacteristic
19C2×#92495C2211yescharacteristic
20C31×#124—21no—
21C31×#124C2211yescharacteristic
22C31×C22⋊Q8—21no—
23C31×C22⋊Q8C2211yescharacteristic
24C31×#121—21no—
25C31×#121C2211yescharacteristic
26C4×Q8×C31C2211yescharacteristic
27C4×D4×C31—21no—
28C4×D4×C31C2211yescharacteristic
29#11758527—24no—
30C2×#92482C2211yescharacteristic
31C2×#92482C2211yescharacteristic
32C2×#92482C2211yescharacteristic
33C2×#92482C2211yescharacteristic
34#11758519—24no—
35C2×#92481C2211yescharacteristic
36C2×#92481C2211yescharacteristic
37#11758515—21no—
38C2×#92480C2211yescharacteristic
39C2×#92480C2211yescharacteristic
40C2×#92479C2211yescharacteristic
41#11758501C2211yescharacteristic
42#11758501C2211yescharacteristic
43C2×#92474C2211yescharacteristic
44C22×C31⋊C8C2211yescharacteristic
45C4⋊C4×D31—21no—
46#11758460—21no—
47#11758460—21no—
48C4×D124C2211yescharacteristic
49#11758446C2211yescharacteristic
50#11758423C2211yescharacteristic
51#11758423C2211yescharacteristic
52#11758420C2211yescharacteristic
53C22⋊(C31⋊C8)C2211yescharacteristic
54C22⋊(C31⋊C8)C2211yescharacteristic
55#11758398C2211yescharacteristic
56#11758398C2211yescharacteristic
57#11758397C2211yescharacteristic
58#11758395C2211yescharacteristic
59C4⋊(C31⋊C8)C2211yescharacteristic
60#11758391C2211yescharacteristic
61#41×C31—22no—
62#41×C31—41no—
63C2×Q8×C31—21no—
64C2×Q8×C31—22no—
65C2×Q8×C31—41no—
66C2×Q8×C31C2311yescharacteristic
67C2×Q8×C31C2311yescharacteristic
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31D411yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31C2311yescharacteristic
75C22×C4×C31C2311yescharacteristic
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78#92495—22no—
79#92495—22no—
80#92495—41no—
81#92495—41no—
82C2×D124C2311yescharacteristic
83C2×C4×D31—21no—
84C2×C4×D31—21no—
85C2×C4×D31—21no—
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31D411yescharacteristic
91C4⋊C4×C31C2311yescharacteristic
92C22⋊C4×C31—21no—
93C22⋊C4×C31—21no—
94C22⋊C4×C31—21no—
95C22⋊C4×C31—21no—
96C22⋊C4×C31D411yescharacteristic
97C22⋊C4×C31D411yescharacteristic
98C42×C31—21no—
99C42×C31C2311yescharacteristic
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
102#92482—22no—
103#92482—22no—
104#92482—22no—
105#92482—41no—
106#92481—22no—
107#92481D414yescomplemented
108#92480—22no—
109#92480—41no—
110#92479—22no—
111(C2×D31)⋊C4—21no—
112C4⋊Dic31—21no—
113C4⋊Dic31—21no—
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117#92474—22no—
118C2×C31⋊C8—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121C2×C31⋊C8C2311yescharacteristic
122C2×C31⋊C8C2311yescharacteristic
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127Q8×C31—21no—
128Q8×C31—21no—
129Q8×C31—21no—
130Q8×C31—42no—
131Q8×C31C2×D412yes—
132D4×C31—21no—
133D4×C31—21no—
134D4×C31—41no—
135D4×C31—42no—
136D4×C31—42no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—42no—
145C2×C4×C31—42no—
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2×D411yescharacteristic
150C2×C4×C31C2411yescharacteristic
151C22⋊D31—42no—
152C22⋊D31—42no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—21no—
156D124—21no—
157D124C2×D412yes—
158C4×D31—42no—
159C4×D31—42no—
160C4×D31—42no—
161Dic62—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62—21no—
165C31⋊C8—21no—
166C31⋊C8—21no—
167C31⋊C8—21no—
168C31⋊C8C2×D412yes—
169#3269—311nomaximal, Sylow
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31C22×D411yescharacteristic
180C2×D31—22no—
181C2×D31—41no—
182C2×D31—41no—
183C4×C31—21no—
184C4×C31—21no—
185C4×C31—22no—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31C22×D411yescharacteristic
191C4×C31H(5,𝔽2)11yescharacteristic
192Dic31—41no—
193Dic31—41no—
194Dic31—41no—
195C2×#137—311no—
196C2×#135—311no—
197#548D3111yescharacteristic, complemented
198#547—311no—
199#497—311no—
200#468—311no—
201#467—311no—
202#463—311no—
203#462—311no—
204#452—311no—
205#452—311no—
206#451—311no—
207#449—311no—
208#440—311no—
209#435—311no—
210C2×C31—41no—
211C2×C31—41no—
212C2×C31—81no—
213C2×C31D4211yescharacteristic
214C2×C31#57511yescharacteristic
215C2×C31#57811yescharacteristic
216D31—42no—
217C2×#41—311no—
218C2×#41—311no—
219C2×#41C2×D3111yescharacteristic
220C22×Q8C2×D3111yescharacteristic
221#137—624no—
222#135—624no—
223C2×Q16—311no—
224C2×Q16—311no—
225C2×Q16—311no—
226C2×Q16—311no—
227C2×SD16—311no—
228C2×SD16—311no—
229C2×SD16—311no—
230C2×SD16—311no—
231C2×D8—311no—
232C2×M16—311no—
233C22×C8—311no—
234C4⋊Q8—311no—
235#124—21no—
236#124C2×D3111yescharacteristic
237#123—621no—
238#123—621no—
239C22⋊Q8—21no—
240C22⋊Q8—311no—
241C22⋊Q8—311no—
242C22⋊Q8C2×D3111yescharacteristic
243#121—21no—
244#121—621no—
245#121C2×D3111yescharacteristic
246C4×Q8C2×D3111yescharacteristic
247C4×D4—21no—
248C4×D4—311no—
249C4×D4C2×D3111yescharacteristic
250C2×C4⋊C4—621no—
251#107—311no—
252C4⋊C8—311no—
253#103—311no—
254#103—311no—
255#103—311no—
256#103—311no—
257#102—311no—
258#102—311no—
259C22⋊C8—311no—
260C22⋊C8—311no—
261#97—311no—
262C31#326911yescharacteristic, complemented, Sylow
263#41—22no—
264#41—41no—
265#41—622no—
266#41—622no—
267#41—1241no—
268#41—1241no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—311no—
273C2×Q8—311no—
274C2×Q8C22×D3111yescharacteristic
275C2×Q8C22×D3111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—311no—
279C2×D4—621no—
280C2×D4—621no—
281C2×D4C22⋊D3111yescharacteristic
282C2×D4C22×D3111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—621no—
286C22×C4—621no—
287C22×C4—621no—
288C22×C4C22×D3111yescharacteristic
289C22×C4C22×D3111yescharacteristic
290Q16—622no—
291Q16—622no—
292Q16—622no—
293Q16—1241no—
294SD16—314no—
295SD16—622no—
296SD16—622no—
297SD16—1241no—
298D8—622no—
299M16—622no—
300C2×C8—311no—
301C2×C8—311no—
302C2×C8—311no—
303C2×C8—311no—
304C2×C8—622no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—311no—
308C4⋊C4—621no—
309C4⋊C4—621no—
310C4⋊C4—621no—
311C4⋊C4—621no—
312C4⋊C4C22⋊D3111yescharacteristic
313C4⋊C4C22×D3111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—621no—
319C22⋊C4—621no—
320C22⋊C4—621no—
321C22⋊C4C22⋊D3111yescharacteristic
322C22⋊C4C22⋊D3111yescharacteristic
323C42—21no—
324C42C22×D3111yescharacteristic
325C23—21no—
326C23—621no—
327C23C2×C22⋊D3111yescharacteristic
328C23C2×C22⋊D3111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—621no—
334Q8—621no—
335Q8—621no—
336Q8—621no—
337Q8D4×D3112yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—312no—
344D4—621no—
345D4—1242no—
346D4—1242no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—621no—
357C2×C4—621no—
358C2×C4—621no—
359C2×C4—1242no—
360C2×C4—1242no—
361C2×C4—1242no—
362C2×C4C2×C22⋊D3111yescharacteristic
363C2×C4C2×C22⋊D3111yescharacteristic
364C2×C4C2×C22⋊D3111yescharacteristic
365C2×C4C2×C22⋊D3111yescharacteristic
366C2×C4C23×D3111yescharacteristic
367C8—312no—
368C8—621no—
369C8—621no—
370C8—621no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—622no—
381C22—1241no—
382C22—1241no—
383C22C22×C22⋊D3111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—1241no—
392C4—1241no—
393C4—1241no—
394C4C2×D4×D3111yescharacteristic
395C4#1175856111yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—1242no—
400C2D4×C22⋊D3111yescharacteristic
401C2#1452184711yescharacteristic
402C2#1452186411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#548C2
socleC22×C31C22×D4
hypercenter#548D31
hypocenterC31#3269
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#11758519C2semidirect
2. C2×#11758527C2semidirect
3. #14521704C2semidirect
4. #14521444C2semidirect
5. #14521293C2semidirect
6. #14521288C2semidirect
7. #14521261C2semidirect
8. #14521259C2semidirect
9. #14521211C2semidirect
10. #14521209C2semidirect
11. #14521205C2semidirect
12. #14521192C2semidirect
13. #14521147C2semidirect
14. #14521121C2semidirect
15. C31×#548C2semidirect
107. #92481D4semidirect
197. #548D31semidirect
262. C31#3269semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
C2×#117585272
#145217042
#145214442
#145212932
#145212882
#145212612
#145212592
#145212112
#145212092
#145212052
#145211922
#145211472
#145211212
C31×#5482
#326931