Finite Group Databasebeta

← #83993

Subgroups — #83993

Order 384.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C211yesmaximal, characteristic
2C2×C3×#137C211yesmaximal, characteristic, complemented
3#5816C211yesmaximal, characteristic, complemented
4C2×#942C211yesmaximal, characteristic, complemented
5#5236C211yesmaximal, characteristic, complemented
6#5205C214yesmaximal, complemented
7#5202C214yesmaximal, complemented
8#5139C211yesmaximal, characteristic, complemented
9#5132C211yesmaximal, characteristic, complemented
10#3022—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic, complemented
13#1010—21no—
14#1010C2212yes—
15#1009—21no—
16#1009C2212yes—
17D3×#41—22no—
18D3×#41—22no—
19#1007—21no—
20#1004—21no—
21#1004—22no—
22C2×#235—22no—
23C2×#235C2211yescharacteristic
24C2×#235C2211yescharacteristic
25C3×#137—22no—
26C3×#137C2214yescomplemented
27C2×C3×Q16—21no—
28C2×C3×SD16—21no—
29C2×C3×M16C2211yescharacteristic, complemented
30#947C2212yescomplemented
31#946—22no—
32#944C2212yescomplemented
33C2×#216—21no—
34#942—22no—
35#942C2214yescomplemented
36C2×#215—21no—
37#929—22no—
38#926C2211yescharacteristic, complemented
39C2×#208C2211yescharacteristic, complemented
40#837C2214yescomplemented
41#825C2214yescomplemented
42#824C2212yescomplemented
43#823C2212yescomplemented
44#585—31no—
45C2×#137—31no—
46C2×#137D311yescharacteristic, complemented
47#536—31no—
48#456—34no—
49#455—34no—
50#421—31no—
51#421—31no—
52#413—31no—
53C3×#41—21no—
54C3×#41—21no—
55C3×#41D412yes—
56C3×#41C2312yes—
57C2×C3×Q8—21no—
58C2×C3×Q8—21no—
59C2×C3×Q8—21no—
60C2×C3×Q8—21no—
61C2×C3×Q8—21no—
62C2×C3×Q8D411yescharacteristic
63C2×C3×Q8C2312yes—
64C2×C3×D4—21no—
65C2×C3×D4D411yescharacteristic
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4C2311yescharacteristic
69C2×#67—24no—
70C2×#67—21no—
71C2×#67D412yes—
72#239—22no—
73#239—22no—
74D3×Q8—22no—
75D3×Q8—22no—
76#237—24no—
77#237—22no—
78#237—22no—
79#237—24no—
80D3×D4—24no—
81D3×D4—22no—
82D3×D4—24no—
83D3×D4—22no—
84#235—24no—
85#235—24no—
86#235—21no—
87#235—21no—
88#235—21no—
89#235—21no—
90#235—21no—
91#235—21no—
92#235—22no—
93#235—22no—
94#235D412yes—
95#235C2312yes—
96C2×D12—22no—
97C2×D12D411yescharacteristic
98C2×D12C2311yescharacteristic
99C2×C4×D3—24no—
100C2×C4×D3—21no—
101C2×C4×D3D412yes—
102C2×Dic6—22no—
103C2×Dic6D411yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×Q16—24no—
106C3×SD16—24no—
107C3×M16—21no—
108C3×M16C2312yes—
109C2×C3×C8—21no—
110#217—21no—
111#216—24no—
112#215—24no—
113(C2×D3)⋊C4—24no—
114C4⋊Dic3—21no—
115Dic3⋊C4—24no—
116C4×Dic3C2312yes—
117#208—21no—
118#208C2312yes—
119C2×C3⋊C8—21no—
1202−1+4—32no—
1212−1+4—61no—
1222−1+4—61no—
123H(5,𝔽2)—32no—
124H(5,𝔽2)—62no—
125H(5,𝔽2)—61no—
126H(5,𝔽2)—61no—
127C2×#41—31no—
128C2×#41—31no—
129C2×#41—62no—
130C2×#41—62no—
131C2×#41—62no—
132C2×#41C2×D311yescharacteristic
133C22×Q8C2×D311yescharacteristic, complemented
134#137—22no—
135#137—34no—
136#137—62no—
137#137C2×D314yescomplemented
138C2×Q16—21no—
139C2×Q16—61no—
140C2×SD16—21no—
141C2×SD16—61no—
142C2×M16—31no—
143C2×M16C2×D311yescharacteristic, complemented
144C4⋊Q8—32no—
145#124—32no—
146#123—62no—
147C22⋊Q8—62no—
148#117—31no—
149#104—34no—
150#104—34no—
151#101—32no—
152#100—32no—
153C23×C3—21no—
154C23×C3C2×D411yescharacteristic
155C22×D3—24no—
156C22×D3—21no—
157C22×D3C2×D412yes—
158C3×Q8—21no—
159C3×Q8—22no—
160C3×Q8—22no—
161C3×Q8—41no—
162C3×Q8—41no—
163C3×Q8C2×D412yes—
164C3×D4—22no—
165C3×D4—22no—
166C3×D4—21no—
167C3×D4C2×D412yes—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180#67—24no—
181#67—24no—
182#67—22no—
183#67—22no—
184#67—22no—
185#67—22no—
186C2×Dic3—24no—
187C2×Dic3—21no—
188C2×Dic3—22no—
189C2×Dic3C2×D412yes—
190D12—24no—
191D12—21no—
192D12—21no—
193D12—21no—
194D12C2×D412yes—
195C4×D3—24no—
196C4×D3—24no—
197C4×D3—22no—
198C4×D3—22no—
199C4×D3—22no—
200C4×D3—22no—
201Dic6—24no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6C2×D412yes—
206C3×C8—22no—
207C3⋊C8—22no—
208#41—21no—
209#41—21no—
210#41—32no—
211#41—32no—
212#41—62no—
213#41—62no—
214#41—64no—
215#41—64no—
216#41—64no—
217#41—64no—
218#41—62no—
219#41—62no—
220#41—62no—
221#41—62no—
222#41—61no—
223#41—61no—
224#41—61no—
225#41—61no—
226#41—61no—
227#41—61no—
228#41#6712yes—
229#41C22×D312yes—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8#6711yescharacteristic
241C2×Q8C22×D312yes—
242C2×D4—21no—
243C2×D4—32no—
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—62no—
247C2×D4—64no—
248C2×D4—64no—
249C2×D4—62no—
250C2×D4—64no—
251C2×D4—61no—
252C2×D4—62no—
253C2×D4#6711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—32no—
257C22×C4—64no—
258C22×C4—61no—
259C22×C4C22×D311yescharacteristic
260Q16—24no—
261Q16—64no—
262SD16—24no—
263SD16—64no—
264M16—21no—
265M16—32no—
266M16—61no—
267M16C22×D312yes—
268C2×C8—21no—
269C2×C8—61no—
270C4⋊C4—64no—
271C4⋊C4—61no—
272C22⋊C4—64no—
273C22⋊C4—61no—
274C42—32no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3#12011yescharacteristic
279C22×C3#12011yescharacteristic
280C22×C3C22×D411yescharacteristic
281C2×D3—22no—
282C2×D3—22no—
283C2×D3—42no—
284C2×D3—81no—
285C2×D3—84no—
286C3×C4—22no—
287C3×C4—42no—
288C3×C4#12011yescharacteristic
289C3×C4#12011yescharacteristic
290C3×C4C22×D411yescharacteristic
291C3×C4C22×D411yescharacteristic
292Dic3—22no—
293Dic3—22no—
294Dic3—42no—
295C23—21no—
296C23—32no—
297C23—64no—
298C23—61no—
299C23D3×D411yescharacteristic
300Q8—21no—
301Q8—22no—
302Q8—22no—
303Q8—32no—
304Q8—41no—
305Q8—41no—
306Q8—64no—
307Q8—61no—
308Q8—61no—
309Q8—61no—
310Q8C2×#6712yes—
311D4—22no—
312D4—22no—
313D4—21no—
314D4—32no—
315D4—64no—
316D4—64no—
317D4—64no—
318D4—62no—
319D4—62no—
320D4—62no—
321D4—62no—
322D4—61no—
323D4—61no—
324D4—61no—
325D4C2×#6712yes—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—32no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—61no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4D3×D411yescharacteristic
343C2×C4D3×D411yescharacteristic
344C2×C4D3×D411yescharacteristic
345C2×C4C2×#6711yescharacteristic
346C2×C4C2×#6711yescharacteristic
347C2×C4C23×D311yescharacteristic
348C8—22no—
349C8—62no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—42no—
354C2×C3C2×#12011yescharacteristic
355D3—42no—
356D3—42no—
357C22—22no—
358C22—41no—
359C22—62no—
360C22—62no—
361C22—81no—
362C22—122no—
363C22—241no—
364C22—244no—
365C22#93711yescharacteristic
366C22#93711yescharacteristic
367C22C2×D3×D411yescharacteristic
368C4—22no—
369C4—42no—
370C4—62no—
371C4—62no—
372C4—122no—
373C4#93711yescharacteristic
374C4#93711yescharacteristic
375C4C2×D3×D411yescharacteristic
376C4C22×#6711yescharacteristic
377C3#302211yescharacteristic, complemented, Sylow
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—122no—
383C2—122no—
384C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC2×C3C2×#120
prime radical#5964C2
hypercenterC2×#137D3
hypocenterC3#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×C3×#137C2semidirect
3. #5816C2semidirect
4. C2×#942C2semidirect
5. #5236C2semidirect
6. #5205C2semidirect
7. #5202C2semidirect
8. #5139C2semidirect
9. #5132C2semidirect
12. C22×C3×Q8C22semidirect
26. C3×#137C22semidirect
29. C2×C3×M16C22semidirect
30. #947C22semidirect
32. #944C22semidirect
35. #942C22semidirect
38. #926C22semidirect
39. C2×#208C22semidirect
40. #837C22semidirect
41. #825C22semidirect
42. #824C22semidirect
43. #823C22semidirect
46. C2×#137D3semidirect
133. C22×Q8C2×D3semidirect
137. #137C2×D3semidirect
143. C2×M16C2×D3semidirect
377. C3#3022semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×C3×#1372
#58162
C2×#9422
#52362
#52052
#52022
#51392
#51322
#30223