Finite Group Databasebeta

← #79797

Subgroups — #79797

Order 384.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#916C211yesmaximal, characteristic, complemented
2C2×#915C211yesmaximal, characteristic, complemented
3#5576C211yesmaximal, characteristic, complemented
4#5555C211yesmaximal, characteristic, complemented
5C3×#440C211yesmaximal, characteristic, complemented
6#5031C211yesmaximal, characteristic, complemented
7#5016C211yesmaximal, characteristic, complemented
8#4887C211yesmaximal, characteristic, complemented
9#4886C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4808C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4781C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3269—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#237C2211yescharacteristic
20C2×#237C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24C2×#216C2211yescharacteristic
25C2×#216C2211yescharacteristic
26C2×#214C2211yescharacteristic
27C2×#213C2211yescharacteristic
28#930—21no—
29C2×C4⋊Dic3—21no—
30#916—24no—
31#915—24no—
32C2×Dic12C2211yescharacteristic
33C2×Dic12C2211yescharacteristic
34C2×#204C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic
38#897—21no—
39#897C2211yescharacteristic
40#897C2211yescharacteristic
41#897C2211yescharacteristic
42#895—21no—
43#895C2211yescharacteristic
44#886—21no—
45#886—21no—
46#875—21no—
47#875C2211yescharacteristic
48C4×D12—21no—
49C4×D12C2211yescharacteristic
50#869C2211yescharacteristic
51C3×#107C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#97C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#810C2211yescharacteristic
58#810C2211yescharacteristic
59#810C2211yescharacteristic
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic
62C2×#137—31no—
63C2×#135—31no—
64#548—31no—
65#547—31no—
66#497—31no—
67#468—31no—
68#467—31no—
69#463—31no—
70#462—31no—
71#452—31no—
72#452—31no—
73#451—31no—
74#449—31no—
75#440D311yescharacteristic, complemented
76#435—31no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84#239—22no—
85#239—41no—
86D3×Q8—22no—
87D3×Q8—41no—
88#237—22no—
89#237—22no—
90#237—41no—
91#237—41no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12D411yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×SD16D414yescomplemented
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C42—21no—
112C3×C42C2311yescharacteristic
113#216—22no—
114#216—22no—
115#214—22no—
116#213—22no—
117(C2×D3)⋊C4—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—
123(C2×D3)⋊C4D411yescharacteristic
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133Dic12—22no—
134Dic12—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×Q16—31no—
147C2×Q16—31no—
148C2×Q16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16C2×D311yescharacteristic
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—31no—
164C22⋊Q8—61no—
165#121—31no—
166#121—61no—
167#121—61no—
168C4×Q8C2×D311yescharacteristic
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C4⋊C4—61no—
173#107C2×D311yescharacteristic
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic
179#102—31no—
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97C2×D311yescharacteristic
184C23×C3—21no—
185C22×D3—21no—
186C22×D3C2×D411yescharacteristic
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8C2×D412yes—
190C3×D4—21no—
191C3×D4C2×D412yes—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2411yescharacteristic
199#67—42no—
200#67—42no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—21no—
209D12—21no—
210D12—41no—
211D12—42no—
212D12—42no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—42no—
218C4×D3—42no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—42no—
226C3×C8—21no—
227C3×C8C2×D412yes—
228C3⋊C8—21no—
229C3⋊C8—21no—
230#41—62no—
231#41—62no—
232#41—62no—
233#41—121no—
234#41—121no—
235#41—121no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—31no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×D4—31no—
244C2×D4—31no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4C22×D311yescharacteristic
250C22×C4—21no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257Q16—62no—
258Q16—62no—
259Q16—62no—
260Q16—121no—
261SD16—62no—
262SD16—62no—
263SD16—121no—
264SD16D1214yescomplemented
265D8—62no—
266M16—62no—
267C2×C8—31no—
268C2×C8—31no—
269C2×C8—62no—
270C2×C8C22×D311yescharacteristic
271C2×C8C22×D311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—31no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4C22×D311yescharacteristic
280C4⋊C4C22×D311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—31no—
283C22⋊C4—31no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C42—21no—
291C42C22×D311yescharacteristic
292C22×C3—22no—
293C22×C3—41no—
294C22×C3—41no—
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—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4C22×D411yescharacteristic
311C3×C4H(5,𝔽2)11yescharacteristic
312Dic3—21no—
313Dic3—21no—
314Dic3—41no—
315Dic3—41no—
316Dic3—41no—
317C23—21no—
318C23—31no—
319C23—31no—
320C23—61no—
321Q8—21no—
322Q8—61no—
323Q8—61no—
324Q8—61no—
325Q8—61no—
326Q8—61no—
327Q8—61no—
328Q8—122no—
329Q8C2×D1212yes—
330D4—21no—
331D4—61no—
332D4—61no—
333D4—121no—
334D4—122no—
335D4—122no—
336D4—122no—
337D4—122no—
338D4C2×D1212yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—42no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4D3×D411yescharacteristic
357C2×C4D3×D411yescharacteristic
358C2×C4C23×D311yescharacteristic
359C8—21no—
360C8—61no—
361C8—61no—
362C8C2×D1212yes—
363C2×C3—42no—
364C2×C3D4211yescharacteristic
365C2×C3#57511yescharacteristic
366C2×C3#57811yescharacteristic
367D3—41no—
368D3—41no—
369D3—81no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—61no—
389C4—61no—
390C4—121no—
391C4—121no—
392C4—121no—
393C4C22×D1211yescharacteristic
394C4#100911yescharacteristic
395C3#326911yescharacteristic, complemented, Sylow
396C2—42no—
397C2—121no—
398C2—121no—
399C2—241no—
400C2D4×D1211yescharacteristic
401C2#577311yescharacteristic
402C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3269
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#916C2semidirect
2. C2×#915C2semidirect
3. #5576C2semidirect
4. #5555C2semidirect
5. C3×#440C2semidirect
6. #5031C2semidirect
7. #5016C2semidirect
8. #4887C2semidirect
9. #4886C2semidirect
10. #4809C2semidirect
11. #4808C2semidirect
12. #4783C2semidirect
13. #4781C2semidirect
14. #4715C2semidirect
15. #4706C2semidirect
75. #440D3semidirect
104. C3×SD16D4semidirect
264. SD16D12semidirect
395. C3#3269semidirect

Maximal subgroups

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

subgroupindex
C2×#9162
C2×#9152
#55762
#55552
C3×#4402
#50312
#50162
#48872
#48862
#48092
#48082
#47832
#47812
#47152
#47062
#32693