Finite Group Databasebeta

← #83464

Subgroups — #83464

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic, complemented
2#5526C211yesmaximal, characteristic, complemented
3#5093C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5060C211yesmaximal, characteristic, complemented
6#5059C211yesmaximal, characteristic, complemented
7#5050C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5040C211yesmaximal, characteristic, complemented
10#5037C211yesmaximal, characteristic
11#5031C211yesmaximal, characteristic, complemented
12#5028C211yesmaximal, characteristic, complemented
13#5019C211yesmaximal, characteristic, complemented
14#5016C211yesmaximal, characteristic, complemented
15#5000C211yesmaximal, characteristic, complemented
16#3225—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×#126—21no—
21C3×#124—21no—
22C3×#124C2211yescharacteristic
23C3×#123—21no—
24C3×C22⋊Q8—21no—
25C3×C22⋊Q8C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic
29C3×#117C2211yescharacteristic
30C2×#216C2211yescharacteristic
31C2×#216C2211yescharacteristic
32C2×#216C2211yescharacteristic
33C2×#215C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic
36C2×#214C2211yescharacteristic
37C2×#213C2211yescharacteristic
38#897—21no—
39#897—21no—
40#886—21no—
41#886—21no—
42#875C2211yescharacteristic
43#875C2211yescharacteristic
44#869C2211yescharacteristic
45#869C2211yescharacteristic
46#830C2211yescharacteristic
47#830C2211yescharacteristic
48#810C2211yescharacteristic
49#810C2211yescharacteristic
50#810C2211yescharacteristic
51#810C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic
57C4⋊(C3⋊C8)C2211yescharacteristic
58#803C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#542D311yescharacteristic, complemented
61#536—31no—
62#497—31no—
63#495—31no—
64#489—31no—
65#487—31no—
66#464—31no—
67#463—31no—
68#462—31no—
69#461—31no—
70#452—31no—
71#451—31no—
72#449—31no—
73#448—31no—
74#433—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C22×Dic3—21no—
83#235—41no—
84#235—41no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×Dic6—21no—
88C2×Dic6—41no—
89C2×Dic6—41no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4D411yescharacteristic
106C3×C22⋊C4D411yescharacteristic
107C3×C42—21no—
108C3×C42C2311yescharacteristic
109C3×C42C2311yescharacteristic
110#216—41no—
111#216—41no—
112#216—41no—
113#215—41no—
114#214—41no—
115#214—41no—
116#214—41no—
117#213—41no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130C2×#41—31no—
131C22×Q8—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8C2×D311yescharacteristic
143#126—21no—
144#124—21no—
145#124—31no—
146#124—31no—
147#124C2×D311yescharacteristic
148#123—21no—
149#123—61no—
150#123—61no—
151C22⋊Q8—21no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8C2×D311yescharacteristic
155#121C2×D311yescharacteristic
156C4×Q8C2×D311yescharacteristic
157C4×D4C2×D311yescharacteristic
158#117C2×D311yescharacteristic
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103—31no—
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—81no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—81no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—81no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199D12—21no—
200D12—21no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—42no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—121no—
214#41—121no—
215C2×Q8—21no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—61no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231Q16—121no—
232Q16—121no—
233Q16—121no—
234SD16—121no—
235SD16—121no—
236SD16—121no—
237SD16—121no—
238D8—121no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—31no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4#6711yescharacteristic
251C4⋊C4#6711yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4#6711yescharacteristic
264C22⋊C4#6711yescharacteristic
265C42—21no—
266C42C22×D311yescharacteristic
267C42C22×D311yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C22×D411yescharacteristic
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4H(5,𝔽2)11yescharacteristic
287C3×C4H(5,𝔽2)11yescharacteristic
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—61no—
293C23C2×#6711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—61no—
297Q8—61no—
298Q8—61no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—81no—
303Q8—122no—
304D4—21no—
305D4—21no—
306D4—61no—
307D4—61no—
308D4—81no—
309D4—122no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—81no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4C2×#6711yescharacteristic
322C2×C4C2×#6711yescharacteristic
323C2×C4C2×#6711yescharacteristic
324C2×C4C2×#6711yescharacteristic
325C2×C4C2×#6711yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—61no—
328C8—61no—
329C8—61no—
330C8—61no—
331C2×C3—41no—
332C2×C3—81no—
333C2×C3#53511yescharacteristic
334C2×C3#57711yescharacteristic
335C2×C3#57811yescharacteristic
336D3—81no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22C22×#6711yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4#100911yescharacteristic
359C4#100911yescharacteristic
360C3#322511yescharacteristic, complemented, Sylow
361C2—41no—
362C2—81no—
363C2—241no—
364C2#583011yescharacteristic
365C2#583611yescharacteristic
366C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#542C2
socleC22×C3C22×D4
prime radical#5037C2
hypercenter#542D3
hypocenterC3#3225
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. C3×#542C2semidirect
2. #5526C2semidirect
3. #5093C2semidirect
4. #5089C2semidirect
5. #5060C2semidirect
6. #5059C2semidirect
7. #5050C2semidirect
8. #5047C2semidirect
9. #5040C2semidirect
11. #5031C2semidirect
12. #5028C2semidirect
13. #5019C2semidirect
14. #5016C2semidirect
15. #5000C2semidirect
60. #542D3semidirect
360. C3#3225semidirect

Maximal subgroups

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

subgroupindex
C3×#5422
#55262
#50932
#50892
#50602
#50592
#50502
#50472
#50402
#50372
#50312
#50282
#50192
#50162
#50002
#32253