Finite Group Databasebeta

← #16855027

Subgroups — #16855027

Order 3968.

712 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758564C211yesmaximal, characteristic, complemented
2C2×#11758495C211yesmaximal, characteristic, complemented
3C2×#11758496C211yesmaximal, characteristic, complemented
4#14521629C211yesmaximal, characteristic, complemented
5#14521210C211yesmaximal, characteristic, complemented
6#14521209C211yesmaximal, characteristic, complemented
7#14521203C211yesmaximal, characteristic, complemented
8#14521197C211yesmaximal, characteristic, complemented
9#14521196C211yesmaximal, characteristic, complemented
10#14521189C211yesmaximal, characteristic, complemented
11#14521188C211yesmaximal, characteristic, complemented
12#14521049C211yesmaximal, characteristic, complemented
13#103×D31C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#451C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17#11758564—24no—
18#11758564—24no—
19C2×#92499—21no—
20C2×#92499C2211yescharacteristic
21C2×#92499C2211yescharacteristic
22C2×#92499C2211yescharacteristic
23C2×Q8×D31—21no—
24C2×Q8×D31C2211yescharacteristic
25C2×Q8×D31C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495—21no—
28C2×#92495C2211yescharacteristic
29C2×#92495C2211yescharacteristic
30C2×Q16×C31C2211yescharacteristic
31C2×Q16×C31C2211yescharacteristic
32C31×C22⋊Q8C2211yescharacteristic
33#11758523C2211yescharacteristic
34C2×#92482C2211yescharacteristic
35C2×#92481C2211yescharacteristic
36C2×#92481C2211yescharacteristic
37C2×#92481C2211yescharacteristic
38#11758507C2211yescharacteristic
39#11758496—24no—
40#11758495—24no—
41C2×D248C2211yescharacteristic
42C2×#92470C2211yescharacteristic
43C2×#92469C2211yescharacteristic
44C2×C8×D31C2211yescharacteristic
45#11758473—21no—
46#11758472—21no—
47#11758472C2211yescharacteristic
48C4⋊C4×D31C2211yescharacteristic
49#11758461—21no—
50#11758461—21no—
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×C22⋊C8C2211yescharacteristic
54#11758423C2211yescharacteristic
55#11758423C2211yescharacteristic
56C22⋊(C31⋊C8)C2211yescharacteristic
57#11758409C2211yescharacteristic
58(C2×D31)⋊C8C2211yescharacteristic
59(C2×D31)⋊C8C2211yescharacteristic
60#11758404C2211yescharacteristic
61#11758398C2211yescharacteristic
62#11758397C2211yescharacteristic
63C2×Q8×C31—22no—
64C2×Q8×C31—22no—
65C2×Q8×C31—21no—
66C2×Q8×C31C2311yescharacteristic
67C2×Q8×C31C2311yescharacteristic
68C2×Q8×C31C2311yescharacteristic
69C22×C4×C31—21no—
70C22×C4×C31C2311yescharacteristic
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31D411yescharacteristic
74C2×C22⋊D31D411yescharacteristic
75#92499—28no—
76#92499—22no—
77#92499—22no—
78#92499—22no—
79#92499—22no—
80#92499—22no—
81#92499D414yescomplemented
82Q8×D31—28no—
83Q8×D31—22no—
84Q8×D31—22no—
85Q8×D31—22no—
86Q8×D31D414yescomplemented
87#92495—28no—
88#92495—28no—
89#92495—22no—
90#92495—22no—
91#92495—22no—
92#92495—22no—
93C2×D124—21no—
94C2×D124—21no—
95C2×D124—21no—
96C2×D124—21no—
97C2×D124D411yescharacteristic
98C2×D124C2311yescharacteristic
99C2×D124C2311yescharacteristic
100C2×C4×D31—21no—
101C2×C4×D31—21no—
102C2×C4×D31—21no—
103C2×C4×D31—21no—
104C2×C4×D31—21no—
105C2×C4×D31—21no—
106C2×C4×D31—21no—
107C2×C4×D31C2311yescharacteristic
108C2×C4×D31C2311yescharacteristic
109C2×Dic62—21no—
110C2×Dic62—21no—
111C2×Dic62D411yescharacteristic
112C2×Dic62C2311yescharacteristic
113Q16×C31—22no—
114Q16×C31—22no—
115C2×C8×C31C2311yescharacteristic
116C2×C8×C31C2311yescharacteristic
117C4⋊C4×C31—21no—
118C4⋊C4×C31C2311yescharacteristic
119C22⋊C4×C31—21no—
120#92482—22no—
121#92481—22no—
122#92481—22no—
123#92481—22no—
124(C2×D31)⋊C4—21no—
125(C2×D31)⋊C4—21no—
126(C2×D31)⋊C4—21no—
127C4⋊Dic31C2311yescharacteristic
128Dic31⋊C4—21no—
129Dic31⋊C4—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132D248—22no—
133#92470—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31C2×D411yescharacteristic
137C22×D31—21no—
138C22×D31—21no—
139C22×D31C2×D411yescharacteristic
140C22×D31C2×D411yescharacteristic
141Q8×C31—24no—
142Q8×C31—21no—
143Q8×C31—21no—
144Q8×C31—21no—
145Q8×C31—21no—
146Q8×C31C2×D412yes—
147Q8×C31C2×D412yes—
148C2×C4×C31—24no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31C2×D411yescharacteristic
156C2×C4×C31C2411yescharacteristic
157C22⋊D31—24no—
158C22⋊D31—22no—
159C22⋊D31—22no—
160C22⋊D31—81no—
161C2×Dic31—21no—
162C2×Dic31—21no—
163C2×Dic31C2×D411yescharacteristic
164C2×Dic31C2×D411yescharacteristic
165D124—24no—
166D124—24no—
167D124—22no—
168D124—21no—
169D124—21no—
170D124—21no—
171D124—42no—
172D124—81no—
173D124C2×D412yes—
174C4×D31—24no—
175C4×D31—24no—
176C4×D31—24no—
177C4×D31—24no—
178C4×D31—21no—
179C4×D31—21no—
180C4×D31—24no—
181C4×D31—42no—
182C4×D31—81no—
183C4×D31C2×D412yes—
184C4×D31C2×D412yes—
185Dic62—24no—
186Dic62—21no—
187Dic62—24no—
188Dic62—22no—
189Dic62C2×D412yes—
190C8×C31—21no—
191C8×C31—21no—
192C31⋊C8—21no—
193C31⋊C8—21no—
194#3013—311nomaximal, Sylow
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31C22×D411yescharacteristic
199C2×D31—21no—
200C2×D31—21no—
201C2×D31—21no—
202C2×D31—22no—
203C2×D31—21no—
204C2×D31—21no—
205C2×D31—41no—
206C2×D31—41no—
207C2×D31—41no—
208C2×D31—41no—
209C2×D31—41no—
210C2×D31#12012yes—
211C4×C31—22no—
212C4×C31—22no—
213C4×C31—21no—
214C4×C31—41no—
215C4×C31—41no—
216C4×C31C22×D411yescharacteristic
217C4×C31C22×D411yescharacteristic
218Dic31—21no—
219Dic31—22no—
220Dic31—41no—
221Dic31#12012yes—
222C2×2−1+4—311no—
223C2×#137—311no—
224C2×#135—311no—
225#537—311no—
226#452—311no—
227#452—311no—
228#451—311no—
229#451D3111yescharacteristic, complemented
230#449—311no—
231#449—311no—
232#448—311no—
233#448—311no—
234#417—311no—
235C2×#103—311no—
236#408—311no—
237C2×C31—41no—
238C2×C31C2×#12011yescharacteristic
239C2×C31#57711yescharacteristic
240C2×C31#57811yescharacteristic
241D31—22no—
242D31—41no—
243D31—42no—
244D31—81no—
2452−1+4—624no—
2462−1+4—624no—
247C2×#41—311no—
248C2×#41—311no—
249C2×#41—311no—
250C2×#41—311no—
251C2×#41—311no—
252C2×#41—621no—
253C2×#41—621no—
254C2×#41—621no—
255C22×Q8—311no—
256C22×Q8—311no—
257C22×Q8—621no—
258C22×Q8C2×D3111yescharacteristic
259#137—624no—
260#135—624no—
261C2×Q16—311no—
262C2×Q16C2×D3111yescharacteristic
263C2×Q16C2×D3111yescharacteristic
264C2×SD16—311no—
265C2×SD16—311no—
266C2×SD16—311no—
267C2×SD16—311no—
268C2×D8—311no—
269C2×M16—311no—
270C22×C8—311no—
271C22⋊Q8—311no—
272C22⋊Q8—621no—
273C22⋊Q8C2×D3111yescharacteristic
274#121—311no—
275#121—311no—
276#121—621no—
277#121—621no—
278#121—621no—
279C2×C4⋊C4—311no—
280#103—311no—
281#103—311no—
282#103—311no—
283#103—311no—
284#103C2×D3111yescharacteristic
285#103C2×D3111yescharacteristic
286#102—311no—
287#102—311no—
288C22⋊C8—311no—
289C22⋊C8—311no—
290C22⋊C8—311no—
291C22⋊C8C2×D3111yescharacteristic
292C31#301311yescharacteristic, complemented, Sylow
293#41—314no—
294#41—622no—
295#41—622no—
296#41—622no—
297#41—622no—
298#41—628no—
299#41—628no—
300#41—622no—
301#41—622no—
302#41—622no—
303#41—622no—
304#41—628no—
305#41—622no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—311no—
310C2×Q8—314no—
311C2×Q8—311no—
312C2×Q8—621no—
313C2×Q8—621no—
314C2×Q8—622no—
315C2×Q8—622no—
316C2×Q8—628no—
317C2×Q8—622no—
318C2×Q8C22×D3111yescharacteristic
319C2×Q8C22×D3111yescharacteristic
320C2×Q8C22×D3111yescharacteristic
321C2×D4—311no—
322C2×D4—311no—
323C2×D4—311no—
324C2×D4—311no—
325C2×D4—311no—
326C2×D4—621no—
327C2×D4—621no—
328C2×D4—621no—
329C2×D4—621no—
330C2×D4—621no—
331C2×D4—621no—
332C22×C4—21no—
333C22×C4—311no—
334C22×C4—311no—
335C22×C4—621no—
336C22×C4—621no—
337C22×C4—621no—
338C22×C4—621no—
339C22×C4—621no—
340C22×C4—621no—
341C22×C4—621no—
342C22×C4C22×D3111yescharacteristic
343Q16—22no—
344Q16—22no—
345Q16—622no—
346SD16—622no—
347SD16—622no—
348SD16—622no—
349SD16—622no—
350D8—622no—
351M16—622no—
352C2×C8—311no—
353C2×C8—311no—
354C2×C8—622no—
355C2×C8C22×D3111yescharacteristic
356C2×C8C22×D3111yescharacteristic
357C4⋊C4—21no—
358C4⋊C4—311no—
359C4⋊C4—621no—
360C4⋊C4—621no—
361C4⋊C4C22×D3111yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—621no—
364C22⋊C4—621no—
365C22⋊C4—621no—
366C23—311no—
367C23—311no—
368C23—621no—
369C23—621no—
370C23D4×D3111yescharacteristic
371Q8—24no—
372Q8—21no—
373Q8—21no—
374Q8—21no—
375Q8—21no—
376Q8—312no—
377Q8—624no—
378Q8—622no—
379Q8—624no—
380Q8—621no—
381Q8D4×D3112yes—
382Q8D4×D3112yes—
383D4—312no—
384D4—622no—
385D4—624no—
386D4—624no—
387D4—622no—
388D4—622no—
389D4—624no—
390D4—621no—
391D4—621no—
392D4—621no—
393D4—1242no—
394D4—2481no—
395D4—2481no—
396C2×C4—24no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—311no—
404C2×C4—312no—
405C2×C4—312no—
406C2×C4—311no—
407C2×C4—624no—
408C2×C4—624no—
409C2×C4—624no—
410C2×C4—624no—
411C2×C4—624no—
412C2×C4—621no—
413C2×C4—621no—
414C2×C4—621no—
415C2×C4—621no—
416C2×C4—1242no—
417C2×C4—2481no—
418C2×C4D4×D3111yescharacteristic
419C2×C4C23×D3111yescharacteristic
420C8—21no—
421C8—21no—
422C8—621no—
423C8—621no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—312no—
428C22—621no—
429C22—621no—
430C22—621no—
431C22—621no—
432C22—622no—
433C22—621no—
434C22—1241no—
435C22—1241no—
436C22—1241no—
437C22—1241no—
438C22—1241no—
439C22C2×D4×D3111yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—312no—
446C4—621no—
447C4—622no—
448C4—1241no—
449C4C2×D4×D3111yescharacteristic
450C4C2×D4×D3111yescharacteristic
451C2—41no—
452C2—622no—
453C2—1241no—
454C2—1242no—
455C2—2481no—
456C2#120×D3111yescharacteristic
457C2#1452185811yescharacteristic
458C2#1452186211yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
hypercenter#451D31
hypocenterC31#3013
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. C2×#11758495C2semidirect
3. C2×#11758496C2semidirect
4. #14521629C2semidirect
5. #14521210C2semidirect
6. #14521209C2semidirect
7. #14521203C2semidirect
8. #14521197C2semidirect
9. #14521196C2semidirect
10. #14521189C2semidirect
11. #14521188C2semidirect
12. #14521049C2semidirect
13. #103×D31C2semidirect
14. #14521014C2semidirect
15. C31×#451C2semidirect
81. #92499D4semidirect
86. Q8×D31D4semidirect
229. #451D31semidirect
292. C31#3013semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#117584952
C2×#117584962
#145216292
#145212102
#145212092
#145212032
#145211972
#145211962
#145211892
#145211882
#145210492
#103×D312
#145210142
C31×#4512
#301331