Finite Group Databasebeta

← #80272

Subgroups — #80272

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
1#5824C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#915C211yesmaximal, characteristic, complemented
4#5633C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5190C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5165C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4795C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4757C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3269—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#237C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#950—24no—
25#947C2211yescharacteristic
26#946C2211yescharacteristic
27#944C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#215C2211yescharacteristic
31#939—21no—
32#935—21no—
33D4×Dic3C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#929—21no—
37C4×#67C2211yescharacteristic
38#924C2211yescharacteristic
39C2×Dic3⋊C4—21no—
40C22×C3⋊C8C2211yescharacteristic
41#915—24no—
42C2×Dic12C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C2×ΓL(1,𝔽25)C2211yescharacteristic
45#896—21no—
46#895C2211yescharacteristic
47#893—21no—
48#887C2211yescharacteristic
49#885—21no—
50#884—21no—
51C3×#103C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic
58#815C2211yescharacteristic
59Dic3⋊C8C2211yescharacteristic
60#810C2211yescharacteristic
61#807C2211yescharacteristic
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#452D311yescharacteristic, complemented
73#451—31no—
74#449—31no—
75#440—31no—
76#435—31no—
77C3×#41—22no—
78C3×#41—41no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89C22×Dic3—21no—
90D3×Q8—22no—
91D3×Q8—41no—
92#237—22no—
93#237—41no—
94#235—22no—
95#235—41no—
96C2×D12—21no—
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3C2311yescharacteristic
101C2×Dic6—21no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×Q16—41no—
105C3×SD16—22no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#217D411yescharacteristic
114#216—22no—
115#216—22no—
116#215—22no—
117#214D414yescomplemented
118#213—22no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4D411yescharacteristic
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41C2×D311yescharacteristic
142C22×Q8—31no—
143#137—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16C2×D311yescharacteristic
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—61no—
164C22⋊Q8C2×D311yescharacteristic
165#121—31no—
166#121—61no—
167#121—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C4⋊C4—61no—
173#107—31no—
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⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×D4—21no—
193C3×D4—42no—
194C3×D4C2×D412yes—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2411yescharacteristic
203#67—41no—
204#67—42no—
205#67—42no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213C2×Dic3C2×D411yescharacteristic
214D12—21no—
215D12—21no—
216D12—42no—
217C4×D3—21no—
218C4×D3—21no—
219C4×D3—42no—
220C4×D3—42no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—42no—
225Dic6C2×D412yes—
226C3×C8—21no—
227C3×C8—21no—
228C3⋊C8—21no—
229C3⋊C8C2×D412yes—
230#41—22no—
231#41—41no—
232#41—62no—
233#41—62no—
234#41—121no—
235#41—121no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8C22×D311yescharacteristic
242C2×Q8C22×D311yescharacteristic
243C2×D4—21no—
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4C22×D311yescharacteristic
250C22×C4—21no—
251C22×C4—31no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4C22×D311yescharacteristic
257Q16—41no—
258Q16—62no—
259Q16—62no—
260Q16—62no—
261SD16—22no—
262SD16—34no—
263SD16—62no—
264SD16—121no—
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—31no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
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—31no—
291C42—61no—
292C22×C3—22no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3C22×D411yescharacteristic
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4C22×D411yescharacteristic
310C3×C4H(5,𝔽2)11yescharacteristic
311Dic3—22no—
312Dic3—21no—
313Dic3—21no—
314Dic3—21no—
315Dic3—41no—
316Dic3—41no—
317C23—21no—
318C23—31no—
319C23—61no—
320C23D3×D411yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—32no—
326Q8—61no—
327Q8—61no—
328Q8—61no—
329Q8—122no—
330D4—21no—
331D4—42no—
332D4—61no—
333D4—61no—
334D4—122no—
335D4—121no—
336D4—122no—
337D4—122no—
338D4D3×D412yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—42no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4D3×D411yescharacteristic
358C2×C4C23×D311yescharacteristic
359C8—21no—
360C8—21no—
361C8—32no—
362C8—61no—
363C2×C3—41no—
364C2×C3—42no—
365C2×C3D4211yescharacteristic
366C2×C3#57511yescharacteristic
367C2×C3#57811yescharacteristic
368D3—41no—
369D3—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—61no—
388C4—61no—
389C4—61no—
390C4—62no—
391C4—121no—
392C4—121no—
393C4C2×D3×D411yescharacteristic
394C4#100411yescharacteristic
395C3#326911yescharacteristic, complemented, Sylow
396C2—41no—
397C2—42no—
398C2—121no—
399C2—241no—
400C2#559011yescharacteristic
401C2#576811yescharacteristic
402C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3269
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #5824C2semidirect
2. C2×#950C2semidirect
3. C2×#915C2semidirect
4. #5633C2semidirect
5. C3×#452C2semidirect
6. #5190C2semidirect
7. #5186C2semidirect
8. #5165C2semidirect
9. #5051C2semidirect
10. #4812C2semidirect
11. #4795C2semidirect
12. #4781C2semidirect
13. #4757C2semidirect
14. #4733C2semidirect
15. #4730C2semidirect
72. #452D3semidirect
117. #214D4semidirect
395. C3#3269semidirect

Maximal subgroups

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

subgroupindex
#58242
C2×#9502
C2×#9152
#56332
C3×#4522
#51902
#51862
#51652
#50512
#48122
#47952
#47812
#47572
#47332
#47302
#32693