Finite Group Databasebeta

← #79901

Subgroups — #79901

Order 384.

448 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
1#5771C211yesmaximal, characteristic, complemented
2#5770C211yesmaximal, characteristic, complemented
3#5753C211yesmaximal, characteristic, complemented
4#5751C211yesmaximal, characteristic, complemented
5C3×#444C211yesmaximal, characteristic, complemented
6#5143C211yesmaximal, characteristic, complemented
7#5141C211yesmaximal, characteristic, complemented
8#4898C211yesmaximal, characteristic, complemented
9#4896C211yesmaximal, characteristic, complemented
10#4827C211yesmaximal, characteristic, complemented
11#4826C211yesmaximal, characteristic, complemented
12#4826C211yesmaximal, characteristic, complemented
13#4825C211yesmaximal, characteristic, complemented
14#4717C211yesmaximal, characteristic, complemented
15#4715C211yesmaximal, characteristic, complemented
16#3289—31nomaximal, Sylow
17#1010C2211yescharacteristic
18#1010C2211yescharacteristic
19#1010C2211yescharacteristic, complemented
20#1009C2211yescharacteristic
21C3×#135C2211yescharacteristic
22C3×#131C2211yescharacteristic
23C3×#131C2211yescharacteristic, complemented
24#951C2211yescharacteristic
25#951C2211yescharacteristic
26#951C2211yescharacteristic, complemented
27#949C2211yescharacteristic
28#918—21no—
29D3×Q16—21no—
30#916—21no—
31#915—21no—
32#915—21no—
33D3×SD16—21no—
34#912—21no—
35#911—21no—
36#909—21no—
37#909—21no—
38#909—21no—
39#909C2211yescharacteristic
40#909C2211yescharacteristic
41#909C2211yescharacteristic, complemented
42#908—21no—
43#908C2211yescharacteristic
44C2×Dic12—21no—
45C2×Dic12C2211yescharacteristic, complemented
46C2×Dic12C2211yescharacteristic, complemented
47#904—21no—
48#904—21no—
49#904C2211yescharacteristic
50C2×#204—21no—
51C2×#204C2211yescharacteristic
52C2×#204C2211yescharacteristic
53#901C2211yescharacteristic
54#901C2211yescharacteristic, complemented
55#875C2211yescharacteristic
56#869C2211yescharacteristic, complemented
57C3×#108C2211yescharacteristic, complemented
58C3×#104C2211yescharacteristic
59C3×#104C2211yescharacteristic, complemented
60C3×#97C2211yescharacteristic, complemented
61#824C2211yescharacteristic, complemented
62#824C2211yescharacteristic, complemented
63#824C2211yescharacteristic, complemented
64#823C2211yescharacteristic
65#819C2211yescharacteristic, complemented
66#805C2211yescharacteristic
67#805C2211yescharacteristic, complemented
68#578—31no—
69#578—31no—
70#577—31no—
71#577—31no—
72#497—31no—
73#473—31no—
74#473—31no—
75#471—31no—
76#471—31no—
77#456—31no—
78#456—31no—
79#455—31no—
80#454—31no—
81#444—31no—
82#444D311yescharacteristic, complemented
83C3×#41C2311yescharacteristic
84C3×#41C2311yescharacteristic
85#239—21no—
86D3×Q8—21no—
87D3×Q8—21no—
88D3×Q8—21no—
89#237—21no—
90#237—21no—
91#237—21no—
92#237—41no—
93#237—41no—
94#237—41no—
95D3×D4—21no—
96D3×D4—41no—
97#235—21no—
98#235—21no—
99#235—21no—
100#235—21no—
101#235C2311yescharacteristic
102#235C2311yescharacteristic
103C2×D12—21no—
104C2×D12C2311yescharacteristic
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6C2311yescharacteristic
109C2×Dic6C2311yescharacteristic
110C2×Dic6C2311yescharacteristic
111C3×Q16D411yescharacteristic
112C3×SD16D411yescharacteristic
113C3×SD16D411yescharacteristic
114C3×D8D411yescharacteristic
115C3×M16—21no—
116C3×M16—21no—
117C3×M16C2311yescharacteristic
118C3×M16C2311yescharacteristic
119C2×C3×C8—21no—
120C2×C3×C8—21no—
121C2×C3×C8C2311yescharacteristic
122C2×C3×C8C2311yescharacteristic
123C3×C42C2311yescharacteristic
124#216—21no—
125#216—21no—
126#216—21no—
127#215—21no—
128#214—21no—
129#214—21no—
130#214—21no—
131#213—21no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—41no—
134#208C2311yescharacteristic
135#208C2311yescharacteristic
136Dic12—21no—
137Dic12—21no—
138Dic12—21no—
139Dic12—21no—
140Dic12—21no—
141Dic12—21no—
142Dic12—21no—
143Dic12D411yescharacteristic
144D24—21no—
145D24D411yescharacteristic
146#204—21no—
147#204—21no—
148#204—21no—
149#204—21no—
150#204—21no—
151#204—21no—
152#204—21no—
153#204—21no—
154#204D411yescharacteristic
155#204D411yescharacteristic
156ΓL(1,𝔽25)—21no—
157ΓL(1,𝔽25)—21no—
158D3×C8—21no—
159D3×C8—21no—
1602−1+4—31no—
1612−1+4—31no—
1622−1+4—31no—
163H(5,𝔽2)—31no—
164#137—31no—
165#137—31no—
166#137—31no—
167#137—31no—
168#137—31no—
169#137—31no—
170#137—61no—
171#137—61no—
172#137—61no—
173#137—61no—
174#137—61no—
175#137—61no—
176AGL(1,ℤ/8)—31no—
177AGL(1,ℤ/8)—31no—
178AGL(1,ℤ/8)—61no—
179AGL(1,ℤ/8)—61no—
180#135—31no—
181#135—61no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135C2×D311yescharacteristic
186C2×Q16—31no—
187C2×Q16—31no—
188C2×Q16—61no—
189C2×Q16—61no—
190C2×SD16—31no—
191C2×SD16—31no—
192C2×SD16—61no—
193C2×SD16—61no—
194#131—31no—
195#131—31no—
196#131C2×D311yescharacteristic
197#131C2×D311yescharacteristic, complemented
198C4⋊Q8—31no—
199#124—31no—
200#108—31no—
201#108C2×D311yescharacteristic, complemented
202#104—31no—
203#104—31no—
204#104C2×D311yescharacteristic
205#104C2×D311yescharacteristic, complemented
206#101—31no—
207#101—31no—
208#101—31no—
209#100—31no—
210#97C2×D311yescharacteristic, complemented
211C22×D3—21no—
212C22×D3—41no—
213C3×Q8C2×D411yescharacteristic
214C3×Q8C2×D411yescharacteristic
215C3×D4—21no—
216C3×D4—21no—
217C3×D4C2×D411yescharacteristic
218C3×D4C2×D411yescharacteristic
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4C2411yescharacteristic
223#67—21no—
224#67—21no—
225#67—41no—
226#67—41no—
227#67—41no—
228#67—41no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235D12—21no—
236D12—21no—
237D12—41no—
238D12C2×D411yescharacteristic
239D12C2×D411yescharacteristic
240C4×D3—21no—
241C4×D3—21no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—41no—
245C4×D3—41no—
246Dic6—21no—
247Dic6—21no—
248Dic6—21no—
249Dic6—21no—
250Dic6—21no—
251Dic6—21no—
252Dic6—41no—
253Dic6—41no—
254Dic6—41no—
255Dic6C2×D411yescharacteristic
256Dic6C2×D411yescharacteristic
257C3×C8—21no—
258C3×C8—21no—
259C3×C8C2×D411yescharacteristic
260C3×C8C2×D411yescharacteristic
261C3×C8C2×D411yescharacteristic
262C3×C8C2×D411yescharacteristic
263C3⋊C8—21no—
264C3⋊C8—21no—
265#41—31no—
266#41—31no—
267#41—61no—
268#41—61no—
269#41—61no—
270#41—61no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41C22×D311yescharacteristic
279#41C22×D311yescharacteristic
280C2×Q8—31no—
281C2×Q8—31no—
282C2×Q8—31no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—31no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—121no—
293Q16—31no—
294Q16—61no—
295Q16—61no—
296Q16—61no—
297Q16—61no—
298Q16—61no—
299Q16—61no—
300Q16—61no—
301Q16—61no—
302Q16—61no—
303Q16—61no—
304Q16D1211yescharacteristic
305SD16—31no—
306SD16—31no—
307SD16—61no—
308SD16—61no—
309SD16—61no—
310SD16—61no—
311SD16—61no—
312SD16—61no—
313SD16—61no—
314SD16—61no—
315SD16—61no—
316SD16—61no—
317SD16—61no—
318SD16—61no—
319SD16D1211yescharacteristic
320SD16D1211yescharacteristic
321D8—31no—
322D8—61no—
323D8—61no—
324D8D1211yescharacteristic
325M16—21no—
326M16—21no—
327M16—31no—
328M16—31no—
329M16—61no—
330M16—61no—
331M16C22×D311yescharacteristic
332M16C22×D311yescharacteristic
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8C22×D311yescharacteristic
338C2×C8C22×D311yescharacteristic
339C4⋊C4—121no—
340C22⋊C4—121no—
341C42C22×D311yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3H(5,𝔽2)11yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—41no—
348C2×D3—81no—
349C2×D3—161no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4C22×D411yescharacteristic
354C3×C4C22×D411yescharacteristic
355Dic3—21no—
356Dic3—21no—
357Dic3—41no—
358Dic3—41no—
359Dic3—41no—
360C23—61no—
361C23—121no—
362Q8—31no—
363Q8—31no—
364Q8—61no—
365Q8—61no—
366Q8—61no—
367Q8—61no—
368Q8—61no—
369Q8—61no—
370Q8—121no—
371Q8—121no—
372Q8—121no—
373Q8C2×D1211yescharacteristic
374Q8C2×D1211yescharacteristic
375D4—21no—
376D4—21no—
377D4—31no—
378D4—31no—
379D4—61no—
380D4—61no—
381D4—61no—
382D4—61no—
383D4—121no—
384D4—121no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4C2×D1211yescharacteristic
389D4C2×D1211yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4C23×D311yescharacteristic
406C8—21no—
407C8—21no—
408C8—61no—
409C8—61no—
410C8C2×D1211yescharacteristic
411C8C2×D1211yescharacteristic
412C8D3×D411yescharacteristic
413C8D3×D411yescharacteristic
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3D4211yescharacteristic
418D3—41no—
419D3—41no—
420D3—81no—
421C22—21no—
422C22—21no—
423C22—61no—
424C22—61no—
425C22—121no—
426C22—241no—
427C22—481no—
428C22#100911yescharacteristic
429C4—21no—
430C4—21no—
431C4—41no—
432C4—61no—
433C4—61no—
434C4—121no—
435C4—121no—
436C4—121no—
437C4C22×D1211yescharacteristic
438C4C2×D3×D411yescharacteristic
439C3#328911yescharacteristic, complemented, Sylow
440C2—21no—
441C2—41no—
442C2—41no—
443C2—121no—
444C2—121no—
445C2—241no—
446C2D4×D1211yescharacteristic
subgroupG/H
centerC2D4×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#444C2
socleC2×C3D42
hypercenter#444D3
hypocenterC3#3289
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #5771C2semidirect
2. #5770C2semidirect
3. #5753C2semidirect
4. #5751C2semidirect
5. C3×#444C2semidirect
6. #5143C2semidirect
7. #5141C2semidirect
8. #4898C2semidirect
9. #4896C2semidirect
10. #4827C2semidirect
11. #4826C2semidirect
12. #4826C2semidirect
13. #4825C2semidirect
14. #4717C2semidirect
15. #4715C2semidirect
19. #1010C22semidirect
23. C3×#131C22semidirect
26. #951C22semidirect
41. #909C22semidirect
45. C2×Dic12C22semidirect
46. C2×Dic12C22semidirect
54. #901C22semidirect
56. #869C22semidirect
57. C3×#108C22semidirect
59. C3×#104C22semidirect
60. C3×#97C22semidirect
61. #824C22semidirect
62. #824C22semidirect
63. #824C22semidirect
65. #819C22semidirect
67. #805C22semidirect
82. #444D3semidirect
197. #131C2×D3semidirect
201. #108C2×D3semidirect
205. #104C2×D3semidirect
210. #97C2×D3semidirect
439. C3#3289semidirect

Maximal subgroups

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

subgroupindex
#57712
#57702
#57532
#57512
C3×#4442
#51432
#51412
#48982
#48962
#48272
#48262
#48252
#47172
#47152
#32893