Finite Group Databasebeta

← #80210

Subgroups — #80210

Order 384.

476 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#5817C211yesmaximal, characteristic, complemented
2C2×#942C211yesmaximal, characteristic, complemented
3C2×#918C211yesmaximal, characteristic, complemented
4#5633C211yesmaximal, characteristic, complemented
5C3×#451C211yesmaximal, characteristic, complemented
6#5191C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5185C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4810C211yesmaximal, characteristic, complemented
12#4798C211yesmaximal, characteristic, complemented
13#4790C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3277—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×Q16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#947—21no—
25#947C2211yescharacteristic
26#947C2211yescharacteristic
27#946—21no—
28#946C2211yescharacteristic
29Q8×Dic3C2211yescharacteristic
30C2×#216C2211yescharacteristic
31C2×#216C2211yescharacteristic
32#942—24no—
33C2×#215C2211yescharacteristic
34C2×#215C2211yescharacteristic
35C2×#215C2211yescharacteristic
36#930—21no—
37#930C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40C2×#208C2211yescharacteristic
41#918—24no—
42C2×D24C2211yescharacteristic
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×#103C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic
58Dic3⋊C8C2211yescharacteristic
59C8×Dic3C2211yescharacteristic
60#809C2211yescharacteristic
61#807C2211yescharacteristic
62C2×#137—31no—
63C2×#137—31no—
64#548—31no—
65#548—31no—
66#495—31no—
67#468—31no—
68#468—31no—
69#464—31no—
70#464—31no—
71#451—31no—
72#451D311yescharacteristic, complemented
73#449—31no—
74#449—31no—
75C4×Q16—31no—
76#436—31no—
77C2×C3×Q8—21no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88#239—22no—
89#239—41no—
90D3×Q8—22no—
91D3×Q8—41no—
92#235—22no—
93#235—41no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×Dic6C2311yescharacteristic
104C3×Q16—22no—
105C3×Q16—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111#217D411yescharacteristic
112#216—22no—
113#216D414yescomplemented
114#215—22no—
115#215—22no—
116#215—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)⋊C4—21no—
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3C2311yescharacteristic
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4D411yescharacteristic
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—41no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143#137—64no—
144#137—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16C2×D311yescharacteristic
148C2×Q16C2×D311yescharacteristic
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—31no—
153C2×D8—31no—
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124—31no—
158#124—61no—
159#124—61no—
160C22⋊Q8—31no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C22⋊Q8C2×D311yescharacteristic
164#121—31no—
165#121—31no—
166#121—61no—
167#121—61no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic
178#103C2×D311yescharacteristic
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183C4×C8—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8—21no—
193C3×Q8—42no—
194C3×Q8C2×D412yes—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—41no—
205#67—42no—
206#67—42no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212D12—21no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—42no—
217D12—42no—
218C4×D3—21no—
219C4×D3—21no—
220C4×D3—42no—
221C4×D3—42no—
222C4×D3—42no—
223Dic6—21no—
224Dic6—42no—
225Dic6C2×D412yes—
226C3×C8—21no—
227C3×C8—21no—
228C3⋊C8—21no—
229C3⋊C8C2×D412yes—
230#41—62no—
231#41—62no—
232#41—121no—
233#41—121no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—31no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8C22×D311yescharacteristic
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C22×C4—21no—
252C22×C4—31no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4C22×D311yescharacteristic
257Q16—22no—
258Q16—34no—
259Q16—41no—
260Q16—62no—
261SD16—62no—
262SD16—62no—
263SD16—62no—
264SD16—62no—
265D8—121no—
266M16—62no—
267M16—62no—
268C2×C8—31no—
269C2×C8—31no—
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⋊C4C22×D311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—31no—
281C22⋊C4—31no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C42—31no—
290C42—61no—
291C42—61no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3C22×D411yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C3×C4—21no—
306C3×C4—22no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4C22×D411yescharacteristic
311C3×C4H(5,𝔽2)11yescharacteristic
312Dic3—22no—
313Dic3—21no—
314Dic3—21no—
315Dic3—21no—
316Dic3—41no—
317C23—31no—
318C23—61no—
319C23—61no—
320C23D3×D411yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—21no—
326Q8—32no—
327Q8—42no—
328Q8—61no—
329Q8—122no—
330Q8D3×D412yes—
331D4—61no—
332D4—61no—
333D4—61no—
334D4—61no—
335D4—122no—
336D4—122no—
337D4—121no—
338D4—122no—
339D4—122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—42no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
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×C3D4211yescharacteristic
365C2×C3#57511yescharacteristic
366C2×C3#57811yescharacteristic
367D3—41no—
368D3—81no—
369D3—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—22no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—61no—
389C4—61no—
390C4—61no—
391C4—62no—
392C4—121no—
393C4C2×D3×D411yescharacteristic
394C4#100411yescharacteristic
395C3#327711yescharacteristic, complemented, Sylow
396C2—41no—
397C2—121no—
398C2—241no—
399C2—241no—
400C2#559011yescharacteristic
401C2#576611yescharacteristic
402C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
hypercenter#451D3
hypocenterC3#3277
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #5817C2semidirect
2. C2×#942C2semidirect
3. C2×#918C2semidirect
4. #5633C2semidirect
5. C3×#451C2semidirect
6. #5191C2semidirect
7. #5186C2semidirect
8. #5185C2semidirect
9. #5049C2semidirect
10. #4812C2semidirect
11. #4810C2semidirect
12. #4798C2semidirect
13. #4790C2semidirect
14. #4734C2semidirect
15. #4729C2semidirect
72. #451D3semidirect
113. #216D4semidirect
395. C3#3277semidirect

Maximal subgroups

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

subgroupindex
#58172
C2×#9422
C2×#9182
#56332
C3×#4512
#51912
#51862
#51852
#50492
#48122
#48102
#47982
#47902
#47342
#47292
#32773