Finite Group Databasebeta

← #79840

Subgroups — #79840

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#919C211yesmaximal, characteristic, complemented
2C2×#918C211yesmaximal, characteristic, complemented
3#5577C211yesmaximal, characteristic, complemented
4#5576C211yesmaximal, characteristic, complemented
5C3×#441C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7#5027C211yesmaximal, characteristic, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4810C211yesmaximal, characteristic, complemented
12#4808C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3271—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#239C2211yescharacteristic
20C2×D3×Q8C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic
24C2×#216C2211yescharacteristic
25C2×#215C2211yescharacteristic
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#919—24no—
29#918—24no—
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic
32C2×#204C2211yescharacteristic
33C2×#204C2211yescharacteristic
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35C2×D3×C8C2211yescharacteristic
36#897—21no—
37#897C2211yescharacteristic
38#895—21no—
39#895—21no—
40#895—21no—
41#895C2211yescharacteristic
42#895C2211yescharacteristic
43#895C2211yescharacteristic
44#875—21no—
45#875C2211yescharacteristic
46#874—21no—
47#874C2211yescharacteristic
48C4×D12—21no—
49C4×D12—21no—
50C4×D12C2211yescharacteristic
51C3×#106C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#97C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59#809C2211yescharacteristic
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic
62C2×#137—31no—
63C2×#135—31no—
64#550—31no—
65#548—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#464—31no—
70#460—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#448—31no—
75#441D311yescharacteristic, complemented
76#435—31no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79#239—22no—
80#239—22no—
81#239—22no—
82#239—41no—
83#239—41no—
84#239—41no—
85D3×Q8—22no—
86D3×Q8—41no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12D411yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×D12C2311yescharacteristic
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3C2311yescharacteristic
104C2×C4×D3C2311yescharacteristic
105C2×Dic6—21no—
106C2×Dic6C2311yescharacteristic
107C3×Q16D414yescomplemented
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C42—21no—
115C3×C42—21no—
116C3×C42C2311yescharacteristic
117#216—22no—
118#215—22no—
119#215—22no—
120#215—22no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133D24—22no—
134D24—41no—
135#204—22no—
136#204—41no—
137ΓL(1,𝔽25)—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×Q8—31no—
143#137—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16C2×D311yescharacteristic
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156#127—31no—
157#127—61no—
158#124—31no—
159#124—61no—
160C22⋊Q8—31no—
161C22⋊Q8—61no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#121—61no—
168C4×Q8C2×D311yescharacteristic
169C4×Q8C2×D311yescharacteristic
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106C2×D311yescharacteristic
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic
177#103C2×D311yescharacteristic
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97C2×D311yescharacteristic
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8C2×D412yes—
192C3×Q8C2×D412yes—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200C2×Dic3—21no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—21no—
209D12—41no—
210D12—42no—
211D12—42no—
212D12—42no—
213D12—42no—
214D12—42no—
215D12—81no—
216D12—81no—
217D12—81no—
218C4×D3—21no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—42no—
223C4×D3—42no—
224C4×D3—42no—
225C4×D3—42no—
226Dic6—21no—
227Dic6—21no—
228Dic6—42no—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
233#41—62no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238#41—121no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263Q16—62no—
264Q16D1214yescomplemented
265SD16—62no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270D8—62no—
271D8—121no—
272M16—62no—
273C2×C8—31no—
274C2×C8—31no—
275C2×C8—62no—
276C2×C8C22×D311yescharacteristic
277C2×C8C22×D311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—31no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—31no—
285C22⋊C4—31no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—21no—
293C42—21no—
294C42C22×D311yescharacteristic
295C22×C3C22×D411yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C3×C4—22no—
312C3×C4—22no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4C22×D411yescharacteristic
318C3×C4H(5,𝔽2)11yescharacteristic
319Dic3—21no—
320Dic3—21no—
321Dic3—41no—
322C23—31no—
323C23—31no—
324C23—61no—
325C23—61no—
326C23—61no—
327Q8—21no—
328Q8—21no—
329Q8—61no—
330Q8—61no—
331Q8—122no—
332Q8C2×D1212yes—
333Q8C2×D1212yes—
334D4—61no—
335D4—61no—
336D4—61no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—121no—
341D4—122no—
342D4—122no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4D3×D411yescharacteristic
365C2×C4D3×D411yescharacteristic
366C2×C4C23×D311yescharacteristic
367C8—21no—
368C8—61no—
369C8—61no—
370C8C2×D1212yes—
371C2×C3D4211yescharacteristic
372C2×C3#57511yescharacteristic
373C2×C3#57711yescharacteristic
374D3—41no—
375D3—41no—
376D3—81no—
377D3—81no—
378D3—81no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—121no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22C2×D3×D411yescharacteristic
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—61no—
402C4—61no—
403C4—121no—
404C4C22×D1211yescharacteristic
405C4#100911yescharacteristic
406C3#327111yescharacteristic, complemented, Sylow
407C2—121no—
408C2—121no—
409C2—241no—
410C2—241no—
411C2—241no—
412C2D4×D1211yescharacteristic
413C2#577711yescharacteristic
414C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#441C2
socleC22×C3C22×D4
hypercenter#441D3
hypocenterC3#3271
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#919C2semidirect
2. C2×#918C2semidirect
3. #5577C2semidirect
4. #5576C2semidirect
5. C3×#441C2semidirect
6. #5028C2semidirect
7. #5027C2semidirect
8. #4865C2semidirect
9. #4864C2semidirect
10. #4810C2semidirect
11. #4810C2semidirect
12. #4808C2semidirect
13. #4806C2semidirect
14. #4712C2semidirect
15. #4706C2semidirect
75. #441D3semidirect
107. C3×Q16D4semidirect
264. Q16D12semidirect
406. C3#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#9192
C2×#9182
#55772
#55762
C3×#4412
#50282
#50272
#48652
#48642
#48102
#48082
#48062
#47122
#47062
#32713