Finite Group Databasebeta

← #80240

Subgroups — #80240

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5833C211yesmaximal, characteristic, complemented
2#5629C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5189C211yesmaximal, characteristic, complemented
6#5187C211yesmaximal, characteristic, complemented
7#5172C211yesmaximal, characteristic, complemented
8#5171C211yesmaximal, characteristic, complemented
9#5043C211yesmaximal, characteristic, complemented
10#4807C211yesmaximal, characteristic, complemented
11#4805C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4777C211yesmaximal, characteristic, complemented
14#4735C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3204—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23#946C2211yescharacteristic
24#946C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#215C2211yescharacteristic
27#939—21no—
28#939—21no—
29#938C2211yescharacteristic
30#938C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930—21no—
33#930C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35#924—21no—
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic
38#896—21no—
39#895—21no—
40#895C2211yescharacteristic
41D3×C4⋊C4C2211yescharacteristic
42#884—21no—
43#884—21no—
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic
53#821C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56#817C2211yescharacteristic
57#817C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#537—31no—
61#537—31no—
62#483—31no—
63#483—31no—
64#481—31no—
65#481—31no—
66#470—31no—
67#469—31no—
68#465—31no—
69#465—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#449—31no—
73#449—31no—
74#408—31no—
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89#239—41no—
90#239—41no—
91#235—41no—
92#235—41no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12D411yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×Dic6D411yescharacteristic
102C3×Q16—41no—
103C3×SD16—41no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109#217—21no—
110#217—21no—
111#215—41no—
112#213—41no—
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
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C2×#41—31no—
132C2×#41C2×D311yescharacteristic
133C2×Q16C2×D311yescharacteristic
134C2×SD16—31no—
135C2×SD16C2×D311yescharacteristic
136C2×D8—31no—
137C2×M16—31no—
138C22×C8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8C2×D311yescharacteristic
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121—61no—
153#121—61no—
154#121—61no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157#106—31no—
158#106—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3C2×D411yescharacteristic
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—42no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—41no—
194#67—41no—
195#67—81no—
196#67—81no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—81no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—41no—
207D12—42no—
208D12—81no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—42no—
214C4×D3—81no—
215Dic6—41no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220#41—41no—
221#41—41no—
222#41—121no—
223#41—121no—
224#41—121no—
225#41—121no—
226C2×Q8—31no—
227C2×Q8C22×D311yescharacteristic
228C2×Q8C22×D311yescharacteristic
229C2×D4—21no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4C22×D311yescharacteristic
240C22×C4—21no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4C22×D311yescharacteristic
247Q16—41no—
248SD16—41no—
249SD16—121no—
250D8—121no—
251M16—121no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—121no—
255C2×C8C22×D311yescharacteristic
256C2×C8C22×D311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—31no—
259C4⋊C4—31no—
260C4⋊C4—31no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4C22×D311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3C22×D411yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297Dic3—21no—
298Dic3—21no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—31no—
304C23—31no—
305C23—61no—
306C23D3×D411yescharacteristic
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—121no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—61no—
316D4—61no—
317D4—121no—
318D4—122no—
319D4—121no—
320D4—121no—
321D4—241no—
322D4—241no—
323D4—241no—
324D4—241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—42no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4D3×D411yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—21no—
346C8—21no—
347C8—61no—
348C8—61no—
349C2×C3—41no—
350C2×C3—81no—
351C2×C3#53411yescharacteristic
352C2×C3#57711yescharacteristic
353C2×C3#57711yescharacteristic
354D3—41no—
355D3—41no—
356D3—81no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—61no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100411yescharacteristic
384C3#320411yescharacteristic, complemented, Sylow
385C2—41no—
386C2—81no—
387C2—121no—
388C2—121no—
389C2—241no—
390C2#558911yescharacteristic
391C2#577011yescharacteristic
392C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3204
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. #5833C2semidirect
2. #5629C2semidirect
3. C3×#452C2semidirect
4. #5238C2semidirect
5. #5189C2semidirect
6. #5187C2semidirect
7. #5172C2semidirect
8. #5171C2semidirect
9. #5043C2semidirect
10. #4807C2semidirect
11. #4805C2semidirect
12. #4782C2semidirect
13. #4777C2semidirect
14. #4735C2semidirect
15. #4727C2semidirect
71. #452D3semidirect
384. C3#3204semidirect

Maximal subgroups

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

subgroupindex
#58332
#56292
C3×#4522
#52382
#51892
#51872
#51722
#51712
#50432
#48072
#48052
#47822
#47772
#47352
#47272
#32043