Finite Group Databasebeta

← #79508

Subgroups — #79508

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×#909C211yesmaximal, characteristic, complemented
2C2×#904C211yesmaximal, characteristic, complemented
3#5553C211yesmaximal, characteristic, complemented
4#5552C211yesmaximal, characteristic, complemented
5C3×#435C211yesmaximal, characteristic, complemented
6#5135C211yesmaximal, characteristic, complemented
7#5114C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4845C211yesmaximal, characteristic, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4734C211yesmaximal, characteristic, complemented
12#4734C211yesmaximal, characteristic, complemented
13#4733C211yesmaximal, characteristic, complemented
14#4716C211yesmaximal, characteristic, complemented
15#4712C211yesmaximal, characteristic, complemented
16#3271—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24#940—21no—
25#939—21no—
26#930C2211yescharacteristic
27#930C2211yescharacteristic
28#930C2211yescharacteristic
29#924C2211yescharacteristic
30#909—24no—
31C2×Dic12C2211yescharacteristic
32C2×Dic12C2211yescharacteristic
33#904—24no—
34C2×D24C2211yescharacteristic
35C2×D24C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×#204C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×#204C2211yescharacteristic
41#896—21no—
42#893—21no—
43#885—21no—
44#884—21no—
45#884—21no—
46#67⋊C4—21no—
47#875C2211yescharacteristic
48#874C2211yescharacteristic
49C4×Dic6C2211yescharacteristic
50C4×Dic6C2211yescharacteristic
51C3×C4⋊C8C2211yescharacteristic
52C3×C22⋊C8C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54C3×#97C2211yescharacteristic
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57#821C2211yescharacteristic
58#817C2211yescharacteristic
59#816C2211yescharacteristic
60#816C2211yescharacteristic
61#816C2211yescharacteristic
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#441—31no—
76#435D311yescharacteristic, complemented
77C2×C3×D4D411yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C22×Dic3—21no—
87#235—22no—
88#235—22no—
89#235—22no—
90#235—41no—
91#235—41no—
92#235—41no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×Dic6—22no—
102C2×Dic6—21no—
103C2×Dic6—41no—
104C2×Dic6C2311yescharacteristic
105C2×Dic6C2311yescharacteristic
106C2×Dic6C2311yescharacteristic
107C3×M16—22no—
108C2×C3×C8—22no—
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4D411yescharacteristic
114C3×C22⋊C4D411yescharacteristic
115C3×C22⋊C4D411yescharacteristic
116C3×C42C2311yescharacteristic
117#217—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123C4⋊Dic3C2311yescharacteristic
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130Dic12—22no—
131Dic12D414yescomplemented
132D24—22no—
133D24—41no—
134#204—22no—
135#204—22no—
136#204—22no—
137#204—22no—
138#204—41no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×Q8—31no—
143#137—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16C2×D311yescharacteristic
155C22×C8C2×D311yescharacteristic
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×Q8—31no—
169C4×Q8—31no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4C2×D311yescharacteristic
173#106—31no—
174C4⋊C8C2×D311yescharacteristic
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8C2×D311yescharacteristic
182C22⋊C8C2×D311yescharacteristic
183#97C2×D311yescharacteristic
184C23×C3C2×D411yescharacteristic
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C3×D4—41no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2411yescharacteristic
199#67—42no—
200#67—42no—
201#67—42no—
202#67—42no—
203#67—81no—
204#67—81no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—42no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—42no—
218D12—81no—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—42no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—42no—
227Dic6C2×D412yes—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8C2×D412yes—
233#41—62no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238#41—121no—
239C2×Q8—31no—
240C2×Q8—31no—
241C2×Q8—31no—
242C2×Q8—62no—
243C2×Q8—61no—
244C2×Q8—121no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4D1211yescharacteristic
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D311yescharacteristic
263Q16—34no—
264Q16—62no—
265SD16—62no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270D8—62no—
271D8—121no—
272M16—22no—
273C2×C8—22no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C2×C8C22×D311yescharacteristic
277C2×C8C22×D311yescharacteristic
278C4⋊C4—31no—
279C4⋊C4—31no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4D1211yescharacteristic
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4D1211yescharacteristic
291C22⋊C4D1211yescharacteristic
292C42—61no—
293C42—61no—
294C42C22×D311yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3C22×D411yescharacteristic
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—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4C22×D411yescharacteristic
316C3×C4H(5,𝔽2)11yescharacteristic
317Dic3—22no—
318Dic3—22no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23C2×D1211yescharacteristic
326C23C2×D1211yescharacteristic
327Q8—32no—
328Q8—32no—
329Q8—61no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—122no—
334D4—41no—
335D4—61no—
336D4—61no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—61no—
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—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4C2×D1211yescharacteristic
363C2×C4C2×D1211yescharacteristic
364C2×C4C2×D1211yescharacteristic
365C2×C4C2×D1211yescharacteristic
366C2×C4C23×D311yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8D3×D412yes—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3D4211yescharacteristic
374C2×C3#57511yescharacteristic
375C2×C3#57711yescharacteristic
376D3—81no—
377D3—81no—
378D3—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—121no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22C22×D1211yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—62no—
400C4—62no—
401C4—121no—
402C4—121no—
403C4—121no—
404C4C2×D3×D411yescharacteristic
405C4#100911yescharacteristic
406C3#327111yescharacteristic, complemented, Sylow
407C2—41no—
408C2—41no—
409C2—241no—
410C2—241no—
411C2—241no—
412C2D4×D1211yescharacteristic
413C2#574811yescharacteristic
414C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3271
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. C2×#904C2semidirect
3. #5553C2semidirect
4. #5552C2semidirect
5. C3×#435C2semidirect
6. #5135C2semidirect
7. #5114C2semidirect
8. #4848C2semidirect
9. #4845C2semidirect
10. #4738C2semidirect
11. #4734C2semidirect
12. #4734C2semidirect
13. #4733C2semidirect
14. #4716C2semidirect
15. #4712C2semidirect
76. #435D3semidirect
131. Dic12D4semidirect
406. C3#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
C2×#9042
#55532
#55522
C3×#4352
#51352
#51142
#48482
#48452
#47382
#47342
#47332
#47162
#47122
#32713