Finite Group Databasebeta

← #84006

Subgroups — #84006

Order 384.

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

normal subgroups only

This group has 698 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
1C2×#1010C211yesmaximal, characteristic, complemented
2C3×#575C211yesmaximal, characteristic, complemented
3#5825C211yesmaximal, characteristic, complemented
4#5822C211yesmaximal, characteristic, complemented
5#5236C211yesmaximal, characteristic, complemented
6#5205C214yesmaximal, complemented
7#5201C214yesmaximal, complemented
8#5139C211yesmaximal, characteristic, complemented
9C2×#824C211yesmaximal, characteristic, complemented
10#3020—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C2×C3×#41C2211yescharacteristic
13#1010—22no—
14#1010—22no—
15#1010—22no—
16#1010C2214yes—
17C2×D3×Q8—21no—
18C2×#237—21no—
19C2×#237—22no—
20C2×#235—21no—
21C2×#235C2211yescharacteristic
22C22×Dic6—21no—
23C22×Dic6C2211yescharacteristic
24C3×#137—21no—
25C3×#137C2212yescomplemented
26C3×AGL(1,ℤ/8)—21no—
27C3×AGL(1,ℤ/8)C2212yescomplemented
28C3×#135—22no—
29C2×C3×M16C2211yescharacteristic
30#951—21no—
31#950—22no—
32#949—21no—
33#944C2212yescomplemented
34#942C2212yescomplemented
35#936C2212yescomplemented
36#935—22no—
37#932C2212yescomplemented
38#926C2211yescharacteristic
39#924—22no—
40C2×#208C2211yescharacteristic
41#837C2214yescomplemented
42#825C2214yescomplemented
43#824C2214yescomplemented
44C2×2−1+4—31no—
45#575—31no—
46#575D311yescharacteristic, complemented
47#536—31no—
48#456—34no—
49#454—34no—
50#421—31no—
51#421—31no—
52C2×#101—31no—
53C3×#41—21no—
54C3×#41—21no—
55C3×#41—21no—
56C3×#41—21no—
57C3×#41—21no—
58C3×#41—21no—
59C3×#41D412yes—
60C3×#41C2312yes—
61C2×C3×Q8D411yescharacteristic
62C2×C3×Q8C2311yescharacteristic
63C2×C3×D4—21no—
64C2×C3×D4—21no—
65C2×C3×D4D411yescharacteristic
66C2×C3×D4C2311yescharacteristic
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4C2311yescharacteristic
70C2×#67—22no—
71C2×#67—21no—
72C22×Dic3—22no—
73C22×Dic3D412yes—
74D3×Q8—24no—
75D3×Q8—24no—
76#237—28no—
77#237—24no—
78#237—24no—
79#237—28no—
80#235—24no—
81#235—21no—
82#235—21no—
83#235—24no—
84#235D412yes—
85#235C2312yes—
86C2×D12D411yescharacteristic
87C2×C4×D3—22no—
88C2×C4×D3—21no—
89C2×Dic6—22no—
90C2×Dic6—24no—
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6—21no—
94C2×Dic6—21no—
95C2×Dic6—24no—
96C2×Dic6D412yes—
97C2×Dic6D411yescharacteristic
98C2×Dic6C2312yes—
99C3×Q16—22no—
100C3×SD16—22no—
101C3×SD16—22no—
102C3×D8—22no—
103C3×M16—21no—
104C3×M16C2312yes—
105C2×C3×C8—21no—
106#217—21no—
107#217—24no—
108#216—22no—
109#215—22no—
110#214—22no—
111#213—22no—
112C4⋊Dic3—21no—
113Dic3⋊C4—24no—
114C4×Dic3C2312yes—
115#208—21no—
116#208C2312yes—
117C2×C3⋊C8—21no—
1182−1+4—34no—
1192−1+4—62no—
1202−1+4—62no—
1212−1+4—62no—
122C2×#41—31no—
123C2×#41—61no—
124C2×#41—62no—
125C2×#41—61no—
126C2×#41C2×D311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C22×Q8—31no—
129C22×Q8—61no—
130C22×Q8—61no—
131#137—21no—
132#137—32no—
133#137—61no—
134#137C2×D312yescomplemented
135AGL(1,ℤ/8)—21no—
136AGL(1,ℤ/8)—32no—
137AGL(1,ℤ/8)—61no—
138AGL(1,ℤ/8)C2×D312yescomplemented
139#135—22no—
140#135—62no—
141C2×M16—31no—
142C2×M16C2×D311yescharacteristic
143C4⋊Q8—32no—
144#124—32no—
145#123—62no—
146C22⋊Q8—62no—
147#117—31no—
148#104—34no—
149#104—34no—
150#101—34no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3C2×D411yescharacteristic
154C22×D3—21no—
155C3×Q8—21no—
156C3×Q8—21no—
157C3×Q8—21no—
158C3×Q8C2×D412yes—
159C3×D4—22no—
160C3×D4—22no—
161C3×D4—21no—
162C3×D4—21no—
163C3×D4—21no—
164C3×D4—41no—
165C3×D4—41no—
166C3×D4C2×D412yes—
167C2×C3×C4—21no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2411yescharacteristic
179#67—28no—
180#67—22no—
181#67—22no—
182C2×Dic3—28no—
183C2×Dic3—22no—
184C2×Dic3—21no—
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3C2×D414yes—
188D12—21no—
189D12C2×D412yes—
190C4×D3—28no—
191C4×D3—22no—
192C4×D3—22no—
193Dic6—28no—
194Dic6—22no—
195Dic6—21no—
196Dic6—22no—
197Dic6—22no—
198Dic6—22no—
199Dic6C2×D412yes—
200C3×C8—22no—
201C3⋊C8—22no—
202#41—21no—
203#41—21no—
204#41—21no—
205#41—21no—
206#41—21no—
207#41—21no—
208#41—32no—
209#41—32no—
210#41—64no—
211#41—64no—
212#41—68no—
213#41—68no—
214#41—64no—
215#41—64no—
216#41—61no—
217#41—61no—
218#41#6712yes—
219#41C22×D312yes—
220C2×Q8—32no—
221C2×Q8—32no—
222C2×Q8—31no—
223C2×Q8—64no—
224C2×Q8—62no—
225C2×Q8—64no—
226C2×Q8—64no—
227C2×Q8—64no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8#6711yescharacteristic
233C2×Q8C22×D311yescharacteristic
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—31no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4#6711yescharacteristic
240C2×D4C22×D311yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—32no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—61no—
247C22×C4C22×D311yescharacteristic
248Q16—22no—
249Q16—62no—
250SD16—22no—
251SD16—22no—
252SD16—62no—
253SD16—62no—
254D8—22no—
255D8—62no—
256M16—21no—
257M16—32no—
258M16—61no—
259M16C22×D312yes—
260C2×C8—21no—
261C2×C8—61no—
262C4⋊C4—64no—
263C4⋊C4—61no—
264C22⋊C4—64no—
265C22⋊C4—61no—
266C42—32no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—41no—
272C22×C3#12011yescharacteristic
273C22×C3#12011yescharacteristic
274C22×C3C22×D411yescharacteristic
275C2×D3—22no—
276C2×D3—42no—
277C3×C4—22no—
278C3×C4—41no—
279C3×C4#12011yescharacteristic
280C3×C4#12011yescharacteristic
281C3×C4C22×D411yescharacteristic
282C3×C4C22×D411yescharacteristic
283Dic3—24no—
284Dic3—22no—
285Dic3—42no—
286C23—21no—
287C23—21no—
288C23—61no—
289C23D3×D411yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—32no—
294Q8—68no—
295Q8—62no—
296Q8—62no—
297Q8—61no—
298Q8—62no—
299Q8—62no—
300Q8C2×#6712yes—
301D4—22no—
302D4—22no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—32no—
307D4—41no—
308D4—41no—
309D4—68no—
310D4—62no—
311D4—62no—
312D4—61no—
313D4C2×#6712yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—34no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—68no—
322C2×C4—68no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—61no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4D3×D411yescharacteristic
330C2×C4D3×D411yescharacteristic
331C2×C4D3×D411yescharacteristic
332C2×C4C2×#6711yescharacteristic
333C2×C4C2×#6711yescharacteristic
334C2×C4C23×D311yescharacteristic
335C8—22no—
336C8—62no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—42no—
341C2×C3—81no—
342C2×C3C2×#12011yescharacteristic
343D3—42no—
344C22—22no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—62no—
350C22—122no—
351C22#93711yescharacteristic
352C22#93711yescharacteristic
353C22C2×D3×D411yescharacteristic
354C4—22no—
355C4—41no—
356C4—64no—
357C4—62no—
358C4—122no—
359C4#93711yescharacteristic
360C4#93711yescharacteristic
361C4C2×D3×D411yescharacteristic
362C4C22×#6711yescharacteristic
363C3#302011yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—122no—
370C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#575C2
socleC2×C3C2×#120
hypercenter#575D3
hypocenterC3#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C3×#575C2semidirect
3. #5825C2semidirect
4. #5822C2semidirect
5. #5236C2semidirect
6. #5205C2semidirect
7. #5201C2semidirect
8. #5139C2semidirect
9. C2×#824C2semidirect
25. C3×#137C22semidirect
27. C3×AGL(1,ℤ/8)C22semidirect
33. #944C22semidirect
34. #942C22semidirect
35. #936C22semidirect
37. #932C22semidirect
41. #837C22semidirect
42. #825C22semidirect
43. #824C22semidirect
46. #575D3semidirect
134. #137C2×D3semidirect
138. AGL(1,ℤ/8)C2×D3semidirect
363. C3#3020semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C3×#5752
#58252
#58222
#52362
#52052
#52012
#51392
C2×#8242
#30203