Finite Group Databasebeta

← #78673

Subgroups — #78673

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5833C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3C3×#408C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5134C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7#5114C211yesmaximal, characteristic, complemented
8#4737C211yesmaximal, characteristic, complemented
9#4736C211yesmaximal, characteristic, complemented
10#4735C211yesmaximal, characteristic, complemented
11#4734C211yesmaximal, characteristic, complemented
12#4734C211yesmaximal, characteristic, complemented
13#4732C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3198—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#235C2211yescharacteristic, complemented
19C22×D12C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic, complemented
22#947—21no—
23#946—21no—
24#940—21no—
25#939—21no—
26#939—21no—
27#938—21no—
28#937—21no—
29#937—21no—
30#930C2211yescharacteristic
31#930C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic
34#930C2211yescharacteristic
35#930C2211yescharacteristic, complemented
36#926C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic, complemented
38#924C2211yescharacteristic
39#924C2211yescharacteristic, complemented
40C2×D24C2211yescharacteristic
41C2×D24C2211yescharacteristic
42C2×#204C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic, complemented
47C3×C22⋊C8C2211yescharacteristic, complemented
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53#821C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#817C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#537—31no—
61#535—31no—
62#485—31no—
63#482—31no—
64#481—31no—
65#480—31no—
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#450—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74#408D311yescharacteristic, complemented
75C23×D3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8D411yescharacteristic
79C2×C3×D4D411yescharacteristic
80C2×C3×D4D411yescharacteristic
81C2×C3×D4D411yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C22×Dic3—21no—
93#235—41no—
94#235—41no—
95C2×D12—21no—
96C2×D12—41no—
97C2×D12—41no—
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3—21no—
102C2×Dic6C2311yescharacteristic
103C3×M16—41no—
104C2×C3×C8—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109#217—21no—
110#217—21no—
111#217—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126D24—41no—
127D24—41no—
128#204—41no—
129#204—41no—
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C22×D4—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×M16C2×D311yescharacteristic, complemented
138C22×C8C2×D311yescharacteristic, complemented
139#127—61no—
140#124—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—31no—
150#121—61no—
151#121—61no—
152#121—61no—
153#120—61no—
154#120—61no—
155#117—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169C22⋊C8C2×D311yescharacteristic
170C22⋊C8C2×D311yescharacteristic
171C22⋊C8C2×D311yescharacteristic, complemented
172C22⋊C8C2×D311yescharacteristic, complemented
173C23×C3C2×D411yescharacteristic
174C23×C3C2×D411yescharacteristic
175C23×C3C2×D411yescharacteristic
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C3×Q8—41no—
183C3×D4—41no—
184C3×D4—41no—
185C3×D4—41no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196#67—42no—
197#67—81no—
198#67—81no—
199#67—81no—
200#67—81no—
201#67—81no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—81no—
209D12—21no—
210D12—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—42no—
216C4×D3—42no—
217Dic6—21no—
218Dic6—21no—
219C3×C8—21no—
220C3×C8—21no—
221C3×C8—21no—
222C3×C8—21no—
223C24—61no—
224#41—41no—
225#41—41no—
226#41—121no—
227#41—121no—
228C2×Q8—31no—
229C2×Q8D1211yescharacteristic
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4D1211yescharacteristic
244C2×D4D1211yescharacteristic
245C2×D4D1211yescharacteristic
246C22×C4—61no—
247C22×C4—61no—
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251SD16—121no—
252SD16—121no—
253D8—121no—
254D8—121no—
255M16—41no—
256C2×C8—41no—
257C2×C8C22×D311yescharacteristic
258C2×C8C22×D311yescharacteristic
259C2×C8C22×D311yescharacteristic
260C2×C8C22×D311yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—31no—
264C4⋊C4—31no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—61no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3C22×D411yescharacteristic
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—84no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4H(5,𝔽2)11yescharacteristic
302C3×C4H(5,𝔽2)11yescharacteristic
303Dic3—41no—
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—122no—
312C23—241no—
313C23—241no—
314C23C2×D1211yescharacteristic
315C23C2×D1211yescharacteristic
316C23C2×D1211yescharacteristic
317Q8—41no—
318Q8—61no—
319Q8—61no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—61no—
324D4—61no—
325D4—61no—
326D4—61no—
327D4—61no—
328D4—61no—
329D4—122no—
330D4—122no—
331D4—241no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4—241no—
350C2×C4C2×D1211yescharacteristic
351C2×C4C2×D1211yescharacteristic
352C2×C4C2×D1211yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3#53411yescharacteristic
362C2×C3#57611yescharacteristic
363C2×C3#57711yescharacteristic
364D3—81no—
365D3—81no—
366D3—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22C22×D1211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—121no—
391C4—121no—
392C4—121no—
393C4—121no—
394C4—121no—
395C4#100411yescharacteristic
396C4#100411yescharacteristic
397C3#319811yescharacteristic, complemented, Sylow
398C2—41no—
399C2—41no—
400C2—41no—
401C2—241no—
402C2—241no—
403C2—241no—
404C2#549311yescharacteristic
405C2#575111yescharacteristic
406C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
hypercenter#408D3
hypocenterC3#3198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #5833C2semidirect
2. #5830C2semidirect
3. C3×#408C2semidirect
4. #5135C2semidirect
5. #5134C2semidirect
6. #5115C2semidirect
7. #5114C2semidirect
8. #4737C2semidirect
9. #4736C2semidirect
10. #4735C2semidirect
11. #4734C2semidirect
12. #4734C2semidirect
13. #4732C2semidirect
14. #4731C2semidirect
15. #4723C2semidirect
18. C2×#235C22semidirect
20. C2×C3×M16C22semidirect
21. C22×C3×C8C22semidirect
35. #930C22semidirect
37. C2×C4⋊Dic3C22semidirect
39. #924C22semidirect
46. C3×C22⋊C8C22semidirect
47. C3×C22⋊C8C22semidirect
52. #821C22semidirect
53. #821C22semidirect
54. #818C22semidirect
55. #818C22semidirect
56. #817C22semidirect
57. #817C22semidirect
58. #816C22semidirect
59. #816C22semidirect
74. #408D3semidirect
137. C2×M16C2×D3semidirect
138. C22×C8C2×D3semidirect
171. C22⋊C8C2×D3semidirect
172. C22⋊C8C2×D3semidirect
397. C3#3198semidirect

Maximal subgroups

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

subgroupindex
#58332
#58302
C3×#4082
#51352
#51342
#51152
#51142
#47372
#47362
#47352
#47342
#47322
#47312
#47232
#31983