Finite Group Databasebeta

← #16855003

Subgroups — #16855003

Order 3968.

700 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758567C211yesmaximal, characteristic
2C2×#11758492C211yesmaximal, characteristic, complemented
3C2×Q16×D31C211yesmaximal, characteristic, complemented
4#14521622C211yesmaximal, characteristic, complemented
5#14521212C211yesmaximal, characteristic, complemented
6#14521211C211yesmaximal, characteristic, complemented
7#14521208C211yesmaximal, characteristic, complemented
8#14521207C211yesmaximal, characteristic, complemented
9#14521206C211yesmaximal, characteristic, complemented
10#14521204C211yesmaximal, characteristic, complemented
11#14521203C211yesmaximal, characteristic, complemented
12#14521056C211yesmaximal, characteristic, complemented
13#14521042C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17#11758567—24no—
18#11758567—24no—
19C2×Q8×D31C2211yescharacteristic
20C2×Q8×D31C2211yescharacteristic
21C2×Q8×D31C2211yescharacteristic
22C2×#92497—21no—
23C2×#92497—21no—
24C2×#92497C2211yescharacteristic
25C2×#92497C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495C2211yescharacteristic
28C22×Dic62—21no—
29C22×Dic62C2211yescharacteristic
30C2×Q16×C31C2211yescharacteristic
31C2×SD16×C31C2211yescharacteristic
32C31×C22⋊Q8C2211yescharacteristic
33#11758523C2211yescharacteristic
34C2×#92482C2211yescharacteristic
35C2×#92482C2211yescharacteristic
36C2×#92482C2211yescharacteristic
37C2×#92480C2211yescharacteristic
38#11758501C2211yescharacteristic
39Q16×D31—24no—
40#11758492—24no—
41C2×Dic124C2211yescharacteristic
42C2×Dic124C2211yescharacteristic
43C2×#92469C2211yescharacteristic
44C2×C8×D31C2211yescharacteristic
45#11758474C2211yescharacteristic
46#11758471—21no—
47#11758469C2211yescharacteristic
48#11758465—21no—
49#11758462—21no—
50#11758460—21no—
51C31×#103C2211yescharacteristic
52C31×#102C2211yescharacteristic
53C31×C22⋊C8C2211yescharacteristic
54#11758423C2211yescharacteristic
55#11758420C2211yescharacteristic
56C22⋊(C31⋊C8)C2211yescharacteristic
57(C2×D31)⋊C8C2211yescharacteristic
58(C2×D31)⋊C8C2211yescharacteristic
59#11758404C2211yescharacteristic
60#11758404C2211yescharacteristic
61#11758398C2211yescharacteristic
62#11758398C2211yescharacteristic
63#41×C31—22no—
64#41×C31—22no—
65C2×Q8×C31C2311yescharacteristic
66C2×Q8×C31C2311yescharacteristic
67C2×D4×C31—21no—
68C2×D4×C31C2311yescharacteristic
69C22×C4×C31—21no—
70C22×C4×C31C2311yescharacteristic
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31D411yescharacteristic
74C2×C22⋊D31D411yescharacteristic
75C22×Dic31—21no—
76C22×Dic31—21no—
77C22×Dic31—21no—
78Q8×D31—22no—
79Q8×D31—22no—
80Q8×D31—22no—
81Q8×D31—22no—
82Q8×D31—22no—
83Q8×D31D414yes—
84#92497—28no—
85#92497—28no—
86#92497—22no—
87#92497—22no—
88#92497—22no—
89#92497D414yes—
90#92495—28no—
91#92495—22no—
92#92495—22no—
93C2×D124D411yescharacteristic
94C2×C4×D31—21no—
95C2×C4×D31—21no—
96C2×C4×D31—21no—
97C2×C4×D31C2311yescharacteristic
98C2×C4×D31C2311yescharacteristic
99C2×Dic62—28no—
100C2×Dic62—21no—
101C2×Dic62—21no—
102C2×Dic62—21no—
103C2×Dic62—22no—
104C2×Dic62—22no—
105C2×Dic62—21no—
106C2×Dic62D411yescharacteristic
107C2×Dic62C2311yescharacteristic
108C2×Dic62C2311yescharacteristic
109C2×Dic62C2311yescharacteristic
110Q16×C31—22no—
111SD16×C31—22no—
112C2×C8×C31C2311yescharacteristic
113C2×C8×C31C2311yescharacteristic
114C4⋊C4×C31—21no—
115C4⋊C4×C31C2311yescharacteristic
116C22⋊C4×C31—21no—
117C22⋊Dic31—21no—
118#92482—22no—
119#92482—22no—
120#92482—22no—
121#92480—22no—
122(C2×D31)⋊C4—21no—
123(C2×D31)⋊C4—21no—
124(C2×D31)⋊C4—21no—
125C4⋊Dic31—21no—
126C4⋊Dic31C2311yescharacteristic
127Dic31⋊C4—21no—
128Dic31⋊C4—21no—
129C4×Dic31—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132Dic124—22no—
133Dic124—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31—21no—
137C23×C31C2×D411yescharacteristic
138C22×D31C2×D411yescharacteristic
139C22×D31C2×D411yescharacteristic
140Q8×C31—21no—
141Q8×C31—21no—
142Q8×C31—21no—
143Q8×C31C2×D412yes—
144D4×C31—24no—
145D4×C31—21no—
146D4×C31C2×D412yes—
147C2×C4×C31—24no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31C2×D411yescharacteristic
154C2×C4×C31C2411yescharacteristic
155C22⋊D31—24no—
156C22⋊D31—22no—
157C22⋊D31—24no—
158C22⋊D31—22no—
159C2×Dic31—24no—
160C2×Dic31—24no—
161C2×Dic31—21no—
162C2×Dic31—21no—
163C2×Dic31—24no—
164C2×Dic31—21no—
165C2×Dic31—21no—
166C2×Dic31C2×D411yescharacteristic
167C2×Dic31C2×D411yescharacteristic
168D124—22no—
169C4×D31—24no—
170C4×D31—24no—
171C4×D31—21no—
172C4×D31—21no—
173C4×D31—42no—
174C4×D31C2×D412yes—
175C4×D31C2×D412yes—
176Dic62—24no—
177Dic62—24no—
178Dic62—21no—
179Dic62—21no—
180Dic62—22no—
181Dic62—24no—
182Dic62—21no—
183Dic62—21no—
184Dic62—42no—
185Dic62C2×D412yes—
186Dic62C2×D412yes—
187C8×C31—21no—
188C8×C31—21no—
189C31⋊C8—21no—
190C31⋊C8—21no—
191#3011—311nomaximal, Sylow
192C22×C31—21no—
193C22×C31—22no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—41no—
197C22×C31—41no—
198C22×C31C22×D411yescharacteristic
199C2×D31—21no—
200C2×D31—21no—
201C2×D31—21no—
202C2×D31—21no—
203C2×D31—21no—
204C2×D31#12012yes—
205C4×C31—21no—
206C4×C31—22no—
207C4×C31—41no—
208C4×C31—41no—
209C4×C31C22×D411yescharacteristic
210C4×C31C22×D411yescharacteristic
211Dic31—21no—
212Dic31—22no—
213Dic31—22no—
214Dic31—41no—
215Dic31—41no—
216Dic31#12012yes—
217C2×2−1+4—311no—
218C2×#137—311no—
219C22×Q16—311no—
220#536—311no—
221#452—311no—
222#452—311no—
223#452—311no—
224#452D3111yescharacteristic, complemented
225#451—311no—
226#451—311no—
227#451—311no—
228#451—311no—
229#419—311no—
230#416—311no—
231#408—311no—
232C2×C31—41no—
233C2×C31—42no—
234C2×C31C2×#12011yescharacteristic
235C2×C31#57811yescharacteristic
236C2×C31#57811yescharacteristic
237D31—22no—
238D31—41no—
2392−1+4—624no—
2402−1+4—624no—
241C2×#41—311no—
242C2×#41—311no—
243C2×#41—311no—
244C2×#41—621no—
245C2×#41—621no—
246C2×#41—621no—
247C2×#41C2×D3111yescharacteristic
248C22×Q8—311no—
249C22×Q8—311no—
250C22×Q8—311no—
251C22×Q8—311no—
252C22×Q8—621no—
253#137—624no—
254C2×Q16—311no—
255C2×Q16—311no—
256C2×Q16—311no—
257C2×Q16—311no—
258C2×Q16—311no—
259C2×Q16—624no—
260C2×Q16C2×D3111yescharacteristic
261C2×SD16—311no—
262C2×SD16C2×D3111yescharacteristic
263C2×M16—311no—
264C22×C8—311no—
265C4⋊Q8—621no—
266#124—621no—
267#123—621no—
268#123—621no—
269C22⋊Q8—311no—
270C22⋊Q8—311no—
271C22⋊Q8—311no—
272C22⋊Q8C2×D3111yescharacteristic
273#117—311no—
274#103—311no—
275#103—311no—
276#103—311no—
277#103—311no—
278#103—311no—
279#103C2×D3111yescharacteristic
280#102—311no—
281#102C2×D3111yescharacteristic
282C22⋊C8—311no—
283C22⋊C8—311no—
284C22⋊C8—311no—
285C22⋊C8C2×D3111yescharacteristic
286C31#301111yescharacteristic, complemented, Sylow
287#41—22no—
288#41—22no—
289#41—314no—
290#41—622no—
291#41—622no—
292#41—628no—
293#41—622no—
294#41—622no—
295#41—628no—
296#41—628no—
297#41—622no—
298C2×Q8—311no—
299C2×Q8—314no—
300C2×Q8—311no—
301C2×Q8—311no—
302C2×Q8—311no—
303C2×Q8—621no—
304C2×Q8—621no—
305C2×Q8—621no—
306C2×Q8—622no—
307C2×Q8—622no—
308C2×Q8—628no—
309C2×Q8—622no—
310C2×Q8—622no—
311C2×Q8—622no—
312C2×Q8—622no—
313C2×Q8—622no—
314C2×Q8—621no—
315C2×Q8C22×D3111yescharacteristic
316C2×Q8C22×D3111yescharacteristic
317C2×D4—21no—
318C2×D4—311no—
319C2×D4—311no—
320C2×D4—311no—
321C2×D4—621no—
322C2×D4—621no—
323C2×D4C22×D3111yescharacteristic
324C22×C4—21no—
325C22×C4—311no—
326C22×C4—311no—
327C22×C4—621no—
328C22×C4—621no—
329C22×C4—621no—
330C22×C4—621no—
331C22×C4—621no—
332C22×C4—621no—
333C22×C4C22×D3111yescharacteristic
334Q16—22no—
335Q16—622no—
336Q16—622no—
337Q16—622no—
338Q16—622no—
339Q16—622no—
340SD16—22no—
341SD16—622no—
342M16—622no—
343C2×C8—311no—
344C2×C8—311no—
345C2×C8—622no—
346C2×C8C22×D3111yescharacteristic
347C2×C8C22×D3111yescharacteristic
348C4⋊C4—21no—
349C4⋊C4—311no—
350C4⋊C4—621no—
351C4⋊C4—621no—
352C4⋊C4—621no—
353C4⋊C4C22×D3111yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—621no—
356C22⋊C4—621no—
357C22⋊C4—621no—
358C22⋊C4—621no—
359C42—621no—
360C23—21no—
361C23—311no—
362C23—311no—
363C23D4×D3111yescharacteristic
364Q8—21no—
365Q8—21no—
366Q8—21no—
367Q8—312no—
368Q8—312no—
369Q8—622no—
370Q8—624no—
371Q8—624no—
372Q8—624no—
373Q8—621no—
374Q8—621no—
375Q8—621no—
376Q8—621no—
377Q8—1242no—
378Q8D4×D3112yes—
379D4—24no—
380D4—21no—
381D4—622no—
382D4—622no—
383D4—622no—
384D4—624no—
385D4—624no—
386D4D4×D3112yes—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—311no—
394C2×C4—312no—
395C2×C4—312no—
396C2×C4—311no—
397C2×C4—624no—
398C2×C4—624no—
399C2×C4—624no—
400C2×C4—624no—
401C2×C4—624no—
402C2×C4—621no—
403C2×C4—621no—
404C2×C4—621no—
405C2×C4—621no—
406C2×C4—621no—
407C2×C4—621no—
408C2×C4—1242no—
409C2×C4D4×D3111yescharacteristic
410C2×C4C23×D3111yescharacteristic
411C8—21no—
412C8—21no—
413C8—621no—
414C8—621no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—312no—
422C22—621no—
423C22—621no—
424C22—621no—
425C22—621no—
426C22—621no—
427C22C2×D4×D3111yescharacteristic
428C4—21no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—312no—
433C4—621no—
434C4—622no—
435C4—622no—
436C4—1241no—
437C4—1241no—
438C4C2×D4×D3111yescharacteristic
439C4C2×D4×D3111yescharacteristic
440C2—41no—
441C2—42no—
442C2—622no—
443C2—1241no—
444C2#120×D3111yescharacteristic
445C2#1452186311yescharacteristic
446C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
prime radicalC2×#11758567C2
hypercenter#452D31
hypocenterC31#3011
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#11758492C2semidirect
3. C2×Q16×D31C2semidirect
4. #14521622C2semidirect
5. #14521212C2semidirect
6. #14521211C2semidirect
7. #14521208C2semidirect
8. #14521207C2semidirect
9. #14521206C2semidirect
10. #14521204C2semidirect
11. #14521203C2semidirect
12. #14521056C2semidirect
13. #14521042C2semidirect
14. #14521014C2semidirect
15. C31×#452C2semidirect
224. #452D31semidirect
286. C31#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
C2×#117584922
C2×Q16×D312
#145216222
#145212122
#145212112
#145212082
#145212072
#145212062
#145212042
#145212032
#145210562
#145210422
#145210142
C31×#4522
#301131