Finite Group Databasebeta

← #80209

Subgroups — #80209

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2C2×#942C211yesmaximal, characteristic, complemented
3C2×D3×Q16C211yesmaximal, characteristic, complemented
4#5633C211yesmaximal, characteristic
5C3×#451C211yesmaximal, characteristic, complemented
6#5190C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5185C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#4807C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4789C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3270—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×Q16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#947C2211yescharacteristic
25#946—21no—
26#946C2211yescharacteristic
27Q8×Dic3C2211yescharacteristic
28#944—21no—
29#944C2211yescharacteristic
30C2×#216C2211yescharacteristic
31C2×#216C2211yescharacteristic
32C2×#216C2211yescharacteristic
33#942—24no—
34C2×#215C2211yescharacteristic
35C2×#215C2211yescharacteristic
36C4×#67—21no—
37C4×#67C2211yescharacteristic
38#924—21no—
39#924C2211yescharacteristic
40C2×#208C2211yescharacteristic
41D3×Q16—24no—
42C2×Dic12C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C2×D3×C8C2211yescharacteristic
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
58#815C2211yescharacteristic
59Dic3⋊C8C2211yescharacteristic
60#810C2211yescharacteristic
61#808C2211yescharacteristic
62C2×#137—31no—
63C22×Q16—31no—
64D4×Q8—31no—
65#548—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#464—31no—
70#462—31no—
71#452—31no—
72#451—31no—
73#451D311yescharacteristic, complemented
74#448—31no—
75#441—31no—
76#435—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×#67D411yescharacteristic
87D3×Q8—22no—
88D3×Q8—22no—
89D3×Q8—41no—
90D3×Q8—41no—
91#235—22no—
92#235—41no—
93C2×D12—21no—
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×Q16—22no—
104C3×Q16—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110#217—21no—
111#217D411yescharacteristic
112#216—22no—
113#216—22no—
114#216D414yescomplemented
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)⋊C4D411yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
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
136Dic12—22no—
137#204—41no—
138D3×C8—22no—
139C2×#41—31no—
140C22×Q8—31no—
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143#137—64no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16—64no—
149C2×Q16C2×D311yescharacteristic
150C2×Q16C2×D311yescharacteristic
151C2×SD16—31no—
152C2×SD16—31no—
153C2×SD16—31no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124—31no—
159#124—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8C2×D311yescharacteristic
166#121—31no—
167#121—61no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic
179#103C2×D311yescharacteristic
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3C2×D411yescharacteristic
187C3×Q8—21no—
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8—42no—
193C3×Q8C2×D412yes—
194C2×C3×C4—21no—
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—22no—
204#67—42no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211D12—21no—
212D12—21no—
213D12—42no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—42no—
217C4×D3—42no—
218C4×D3—42no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
222Dic6—42no—
223Dic6—42no—
224Dic6C2×D412yes—
225C3×C8—21no—
226C3×C8—21no—
227C3⋊C8—21no—
228C3⋊C8C2×D412yes—
229#41—62no—
230#41—121no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—31no—
234C2×Q8—31no—
235C2×Q8—41no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4—61no—
249C22×C4—21no—
250C22×C4—31no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C22×D311yescharacteristic
255Q16—22no—
256Q16—34no—
257Q16—41no—
258Q16—62no—
259Q16—62no—
260Q16—62no—
261SD16—62no—
262SD16—62no—
263SD16—121no—
264M16—62no—
265C2×C8—31no—
266C2×C8—31no—
267C2×C8—62no—
268C2×C8C22×D311yescharacteristic
269C2×C8C22×D311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4C22×D311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—31no—
282C22⋊C4—31no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C42—31no—
289C42—61no—
290C42—61no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3C22×D411yescharacteristic
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3#13712yescomplemented
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4C22×D411yescharacteristic
307C3×C4H(5,𝔽2)11yescharacteristic
308Dic3—22no—
309Dic3—21no—
310Dic3—21no—
311Dic3—21no—
312Dic3—41no—
313Dic3—41no—
314C23—31no—
315C23—61no—
316C23D3×D411yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—32no—
323Q8—42no—
324Q8—61no—
325Q8—61no—
326Q8—61no—
327Q8—122no—
328Q8—122no—
329Q8D3×D412yes—
330D4—62no—
331D4—61no—
332D4—61no—
333D4—122no—
334D4—122no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—31no—
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—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4D3×D411yescharacteristic
354C2×C4C23×D311yescharacteristic
355C8—21no—
356C8—21no—
357C8—32no—
358C8—61no—
359C2×C3—41no—
360C2×C3D4211yescharacteristic
361C2×C3C2×#13711yescharacteristic
362C2×C3#57811yescharacteristic
363D3—22no—
364D3—81no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—32no—
369C22—61no—
370C22—61no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—61no—
381C4—61no—
382C4—61no—
383C4—62no—
384C4—121no—
385C4—121no—
386C4C2×D3×D411yescharacteristic
387C4#100411yescharacteristic
388C3#327011yescharacteristic, complemented, Sylow
389C2—41no—
390C2—62no—
391C2—241no—
392C2#559011yescharacteristic
393C2#576611yescharacteristic
394C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radical#5633C2
hypercenter#451D3
hypocenterC3#3270
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. Q8×#67C2semidirect
2. C2×#942C2semidirect
3. C2×D3×Q16C2semidirect
5. C3×#451C2semidirect
6. #5190C2semidirect
7. #5186C2semidirect
8. #5185C2semidirect
9. #5052C2semidirect
10. #4807C2semidirect
11. #4806C2semidirect
12. #4795C2semidirect
13. #4789C2semidirect
14. #4733C2semidirect
15. #4728C2semidirect
73. #451D3semidirect
114. #216D4semidirect
300. C2×D3#137semidirect
388. C3#3270semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
C2×#9422
C2×D3×Q162
#56332
C3×#4512
#51902
#51862
#51852
#50522
#48072
#48062
#47952
#47892
#47332
#47282
#32703