Finite Group Databasebeta

← #78996

Subgroups — #78996

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
2#5822C211yesmaximal, characteristic, complemented
3#5748C211yesmaximal, characteristic, complemented
4#5526C211yesmaximal, characteristic, complemented
5C3×#421C211yesmaximal, characteristic, complemented
6C2×#824C211yesmaximal, characteristic, complemented
7#5005C211yesmaximal, characteristic, complemented
8#4827C214yesmaximal, complemented
9#4826C214yesmaximal, complemented
10#3020—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1010—22no—
13#1010—22no—
14#1010—22no—
15#1010C2214yes—
16C2×D3×Q8—21no—
17C2×#237—21no—
18C2×#237—22no—
19C2×#235—21no—
20C2×#235C2211yescharacteristic
21C2×#235C2211yescharacteristic
22C22×Dic6—21no—
23C22×Dic6C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic
25C3×#117C2211yescharacteristic
26#951C2212yescomplemented
27#950—22no—
28#949C2212yescomplemented
29#942—21no—
30#932—21no—
31C2×#208C2211yescharacteristic
32#909—21no—
33#909C2212yescomplemented
34#908—21no—
35#908C2212yescomplemented
36#904—22no—
37#897—22no—
38#886—22no—
39#875C2212yescomplemented
40#869C2212yescomplemented
41C3×#104C2214yescomplemented
42#824C2214yescomplemented
43#805C2214yescomplemented
44C2×2−1+4—31no—
45#575—31no—
46#575—31no—
47#536—31no—
48#456—34no—
49#454—34no—
50#421—31no—
51#421D311yescharacteristic, complemented
52C2×#101—31no—
53C3×#41—21no—
54C3×#41—21no—
55C3×#41D412yes—
56C3×#41C2312yes—
57C2×C3×Q8D411yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4D411yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—21no—
64C2×#67—21no—
65C22×Dic3—22no—
66C22×Dic3D412yes—
67D3×Q8—24no—
68D3×Q8—24no—
69#237—28no—
70#237—24no—
71#237—24no—
72#237—28no—
73#235—24no—
74#235—21no—
75#235—21no—
76#235—24no—
77#235—21no—
78#235—21no—
79#235—21no—
80#235—21no—
81#235D412yes—
82#235C2312yes—
83C2×D12D411yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—22no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×Dic6—22no—
89C2×Dic6—24no—
90C2×Dic6—21no—
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6—21no—
94C2×Dic6—24no—
95C2×Dic6D412yes—
96C2×Dic6D411yescharacteristic
97C2×Dic6C2311yescharacteristic
98C2×Dic6C2312yes—
99C3×M16—21no—
100C3×M16C2312yes—
101C2×C3×C8—21no—
102C3×C4⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42C2312yes—
105#216—22no—
106#215—22no—
107#214—22no—
108#213—22no—
109(C2×D3)⋊C4—24no—
110C4⋊Dic3—24no—
111#208—21no—
112#208C2312yes—
113C2×C3⋊C8—21no—
114Dic12—22no—
115D24—22no—
116#204—22no—
117#204—22no—
1182−1+4—34no—
1192−1+4—62no—
1202−1+4—62no—
1212−1+4—62no—
122C2×#41—31no—
123C2×#41—31no—
124C2×#41—61no—
125C2×#41—62no—
126C2×#41—61no—
127C2×#41C2×D311yescharacteristic
128C22×Q8—31no—
129C22×Q8—61no—
130C22×Q8—61no—
131#137—32no—
132#137—32no—
133#137—61no—
134#137—61no—
135AGL(1,ℤ/8)—32no—
136AGL(1,ℤ/8)—32no—
137AGL(1,ℤ/8)—61no—
138AGL(1,ℤ/8)—61no—
139#135—62no—
140#135—62no—
141C2×M16—31no—
142C2×M16C2×D311yescharacteristic
143C4⋊Q8—32no—
144#124—32no—
145#123—62no—
146C22⋊Q8—62no—
147#117C2×D311yescharacteristic
148#104—34no—
149#104C2×D314yescomplemented
150#101—34no—
151C23×C3—21no—
152C23×C3C2×D411yescharacteristic
153C22×D3—21no—
154C22×D3—21no—
155C3×Q8—21no—
156C3×Q8C2×D412yes—
157C3×D4—22no—
158C3×D4—22no—
159C3×D4—21no—
160C3×D4C2×D412yes—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4C2×D411yescharacteristic
166C2×C3×C4C2×D411yescharacteristic
167C2×C3×C4C2×D411yescharacteristic
168C2×C3×C4C2×D411yescharacteristic
169C2×C3×C4C2×D411yescharacteristic
170C2×C3×C4C2411yescharacteristic
171#67—28no—
172#67—22no—
173#67—22no—
174#67—41no—
175#67—41no—
176C2×Dic3—28no—
177C2×Dic3—22no—
178C2×Dic3—21no—
179C2×Dic3—22no—
180C2×Dic3—21no—
181C2×Dic3C2×D414yes—
182D12—21no—
183D12—21no—
184D12—21no—
185D12C2×D412yes—
186C4×D3—28no—
187C4×D3—22no—
188C4×D3—22no—
189C4×D3—41no—
190C4×D3—41no—
191Dic6—28no—
192Dic6—22no—
193Dic6—21no—
194Dic6—22no—
195Dic6—22no—
196Dic6—22no—
197Dic6—21no—
198Dic6—21no—
199Dic6C2×D412yes—
200C3×C8—22no—
201C3⋊C8—22no—
202#41—21no—
203#41—21no—
204#41—32no—
205#41—32no—
206#41—64no—
207#41—64no—
208#41—68no—
209#41—68no—
210#41—64no—
211#41—64no—
212#41—61no—
213#41—61no—
214#41—61no—
215#41—61no—
216#41—61no—
217#41—61no—
218#41D1212yes—
219#41C22×D312yes—
220C2×Q8—32no—
221C2×Q8—32no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—64no—
225C2×Q8—62no—
226C2×Q8—64no—
227C2×Q8—64no—
228C2×Q8—64no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8D1211yescharacteristic
234C2×D4—21no—
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4D1211yescharacteristic
241C22×C4—21no—
242C22×C4—32no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4C22×D311yescharacteristic
248Q16—62no—
249Q16—62no—
250SD16—62no—
251SD16—62no—
252SD16—62no—
253SD16—62no—
254D8—62no—
255D8—62no—
256M16—21no—
257M16—32no—
258M16—61no—
259M16C22×D312yes—
260C2×C8—21no—
261C2×C8—61no—
262C4⋊C4—21no—
263C4⋊C4—64no—
264C22⋊C4—21no—
265C22⋊C4—64no—
266C42C22×D312yes—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3—42no—
270C22×C3#12011yescharacteristic
271C22×C3#12011yescharacteristic
272C22×C3C22×D411yescharacteristic
273C2×D3—22no—
274C2×D3—42no—
275C2×D3—42no—
276C2×D3—41no—
277C3×C4—22no—
278C3×C4—42no—
279C3×C4#12011yescharacteristic
280C3×C4#12011yescharacteristic
281C3×C4C22×D411yescharacteristic
282C3×C4C22×D411yescharacteristic
283Dic3—24no—
284Dic3—22no—
285Dic3—41no—
286C23—21no—
287C23—61no—
288C23—61no—
289C23D3×D411yescharacteristic
290Q8—21no—
291Q8—32no—
292Q8—68no—
293Q8—62no—
294Q8—62no—
295Q8—61no—
296Q8—62no—
297Q8—62no—
298Q8—61no—
299Q8—61no—
300Q8C2×D1212yes—
301D4—22no—
302D4—22no—
303D4—21no—
304D4—32no—
305D4—68no—
306D4—62no—
307D4—62no—
308D4—61no—
309D4—61no—
310D4—61no—
311D4—121no—
312D4—121no—
313D4C2×D1212yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—34no—
319C2×C4—68no—
320C2×C4—68no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—61no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4C2×D1211yescharacteristic
330C2×C4C2×D1211yescharacteristic
331C2×C4D3×D411yescharacteristic
332C2×C4D3×D411yescharacteristic
333C2×C4D3×D411yescharacteristic
334C2×C4C23×D311yescharacteristic
335C8—22no—
336C8—62no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—42no—
341C2×C3C2×#12011yescharacteristic
342D3—42no—
343D3—81no—
344C22—22no—
345C22—41no—
346C22—42no—
347C22—62no—
348C22—122no—
349C22—122no—
350C22—121no—
351C22#88211yescharacteristic
352C22#88211yescharacteristic
353C22C2×D3×D411yescharacteristic
354C4—22no—
355C4—42no—
356C4—64no—
357C4—62no—
358C4—121no—
359C4#88211yescharacteristic
360C4#88211yescharacteristic
361C4C22×D1211yescharacteristic
362C4C2×D3×D411yescharacteristic
363C3#302011yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—122no—
369C2—241no—
370C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#421C2
socleC2×C3C2×#120
hypercenter#421D3
hypocenterC3#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. #5822C2semidirect
3. #5748C2semidirect
4. #5526C2semidirect
5. C3×#421C2semidirect
6. C2×#824C2semidirect
7. #5005C2semidirect
8. #4827C2semidirect
9. #4826C2semidirect
26. #951C22semidirect
28. #949C22semidirect
33. #909C22semidirect
35. #908C22semidirect
39. #875C22semidirect
40. #869C22semidirect
41. C3×#104C22semidirect
42. #824C22semidirect
43. #805C22semidirect
51. #421D3semidirect
149. #104C2×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
#58222
#57482
#55262
C3×#4212
C2×#8242
#50052
#48272
#48262
#30203