Finite Group Databasebeta

← #80211

Subgroups — #80211

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic
2C22×#216C211yesmaximal, characteristic, complemented
3C2×D3×Q16C211yesmaximal, characteristic, complemented
4#5632C211yesmaximal, characteristic
5C3×#451C211yesmaximal, characteristic, complemented
6#5186C211yesmaximal, characteristic, complemented
7#5183C211yesmaximal, characteristic, complemented
8#5182C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#4813C211yesmaximal, characteristic, complemented
11#4809C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4790C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic
16D4×Q16—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×Q16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#946—21no—
25#946C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic
27#944—21no—
28#944C2211yescharacteristic
29#944C2211yescharacteristic
30C2×#216—24no—
31C2×#216C2211yescharacteristic
32C2×#216C2211yescharacteristic
33C2×#216C2211yescharacteristic
34C2×#216C2211yescharacteristic
35C2×#216C2211yescharacteristic
36C4×#67—21no—
37C4×#67C2211yescharacteristic
38#924—21no—
39#924C2211yescharacteristic
40C22×C3⋊C8C2211yescharacteristic
41D3×Q16—24no—
42C2×Dic12C2211yescharacteristic
43C2×Dic12C2211yescharacteristic
44C2×D3×C8C2211yescharacteristic
45#897C2211yescharacteristic
46#896—21no—
47#888—21no—
48#887C2211yescharacteristic
49#67⋊C4—21no—
50#878—21no—
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58Dic3⋊C8C2211yescharacteristic
59C8×Dic3C2211yescharacteristic
60#810C2211yescharacteristic
61#807C2211yescharacteristic
62C22×Q16—31no—
63C22×Q16—31no—
64D4×Q8—31no—
65D4×Q8—31no—
66#494—31no—
67#467—31no—
68#467—31no—
69#462—31no—
70#462—31no—
71#451—31no—
72#451—31no—
73#451—31no—
74#451D311yescharacteristic, complemented
75C4×Q16—31no—
76C8×D4—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×#67D411yescharacteristic
86C22×Dic3—21no—
87D3×Q8—22no—
88D3×Q8—22no—
89D3×Q8—41no—
90D3×Q8—41no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—22no—
95C2×Dic6—21no—
96C2×Dic6—21no—
97C2×Dic6—21no—
98C2×Dic6—41no—
99C2×Dic6C2311yescharacteristic
100C2×Dic6C2311yescharacteristic
101C2×Dic6C2311yescharacteristic
102C3×Q16—22no—
103C3×Q16—41no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109#217—21no—
110#217D411yescharacteristic
111#216—22no—
112#216—22no—
113#216—22no—
114#216—22no—
115#216D414yescomplemented
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4D411yescharacteristic
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
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
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—22no—
137Dic12—41no—
138D3×C8—22no—
139C22×Q8—31no—
140C22×Q8—31no—
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143C2×Q16—31no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16—31no—
149C2×Q16—31no—
150C2×Q16—64no—
151C2×Q16—64no—
152C2×Q16C2×D311yescharacteristic
153C2×Q16C2×D311yescharacteristic
154C22×C8—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8C2×D311yescharacteristic
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103—31no—
179#103C2×D311yescharacteristic
180#103C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183C4×C8—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3C2×D411yescharacteristic
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—42no—
192C3×Q8C2×D412yes—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67Q1614yescomplemented
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—42no—
214C4×D3—42no—
215Dic6—21no—
216Dic6—21no—
217Dic6—21no—
218Dic6—21no—
219Dic6—21no—
220Dic6—42no—
221Dic6—42no—
222Dic6—42no—
223Dic6C2×D412yes—
224C3×C8—21no—
225C3×C8—21no—
226C3⋊C8—21no—
227C3⋊C8C2×D412yes—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—31no—
233C2×Q8—41no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—62no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×Q8C22×D311yescharacteristic
246C2×D4—31no—
247C22×C4—21no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4C22×D311yescharacteristic
253Q16—22no—
254Q16—34no—
255Q16—41no—
256Q16—62no—
257Q16—62no—
258Q16—62no—
259Q16—62no—
260Q16—62no—
261Q16—121no—
262C2×C8—31no—
263C2×C8—31no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8C22×D311yescharacteristic
267C2×C8C22×D311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4C22×D311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—31no—
283C22⋊C4—31no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C42—31no—
288C42—61no—
289C42—61no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3C2×Q1612yescomplemented
293C22×C3C22×D411yescharacteristic
294C2×D3—21no—
295C2×D3—21no—
296C2×D3C2×Q1612yescomplemented
297C3×C4—21no—
298C3×C4—22no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4C22×D411yescharacteristic
303C3×C4H(5,𝔽2)11yescharacteristic
304Dic3—22no—
305Dic3—21no—
306Dic3—21no—
307Dic3—41no—
308Dic3—41no—
309Dic3—41no—
310Dic3C2×Q1612yescomplemented
311C23—31no—
312C23D3×D411yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—32no—
319Q8—42no—
320Q8—61no—
321Q8—61no—
322Q8—61no—
323Q8—61no—
324Q8—61no—
325Q8—122no—
326Q8—122no—
327Q8—122no—
328Q8D3×D412yes—
329D4—34no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—42no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4D3×D411yescharacteristic
350C2×C4C23×D311yescharacteristic
351C8—21no—
352C8—21no—
353C8—32no—
354C8—61no—
355C2×C3—22no—
356C2×C3D4211yescharacteristic
357C2×C3C22×Q1611yescharacteristic
358C2×C3#57811yescharacteristic
359D3—22no—
360C22—21no—
361C22—21no—
362C22—32no—
363C22—61no—
364C22—61no—
365C22D3×Q1612yescomplemented
366C22C2×D3×D411yescharacteristic
367C4—21no—
368C4—22no—
369C4—32no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—61no—
374C4—61no—
375C4—62no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4C2×D3×D411yescharacteristic
380C4#100411yescharacteristic
381C3D4×Q1611yescharacteristic, complemented, Sylow
382C2—22no—
383C2—62no—
384C2#559011yescharacteristic
385C2C2×D3×Q1611yescharacteristic
386C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#451D3
hypocenterC3D4×Q16
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
2. C22×#216C2semidirect
3. C2×D3×Q16C2semidirect
5. C3×#451C2semidirect
6. #5186C2semidirect
7. #5183C2semidirect
8. #5182C2semidirect
9. #5051C2semidirect
10. #4813C2semidirect
11. #4809C2semidirect
12. #4795C2semidirect
13. #4790C2semidirect
14. #4739C2semidirect
74. #451D3semidirect
115. #216D4semidirect
202. #67Q16semidirect
292. C22×C3C2×Q16semidirect
296. C2×D3C2×Q16semidirect
310. Dic3C2×Q16semidirect
365. C22D3×Q16semidirect
381. C3D4×Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
C22×#2162
C2×D3×Q162
#56322
C3×#4512
#51862
#51832
#51822
#50512
#48132
#48092
#47952
#47902
#47392
#67⋊C82
D4×Q163