Finite Group Databasebeta

← #80052

Subgroups — #80052

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2#5618C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic, complemented
4C2×#835C211yesmaximal, characteristic, complemented
5#5174C211yesmaximal, characteristic, complemented
6#5170C211yesmaximal, characteristic, complemented
7#5164C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4814C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4797C211yesmaximal, characteristic, complemented
13#4796C211yesmaximal, characteristic, complemented
14#4730C211yesmaximal, characteristic, complemented
15#4722C211yesmaximal, characteristic, complemented
16#3320—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#946—21no—
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25#944—21no—
26#944C2211yescharacteristic
27#939—21no—
28#938C2211yescharacteristic
29D4×Dic3—21no—
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35#924—21no—
36#924C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39#892C2211yescharacteristic
40#890C2211yescharacteristic
41#886—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#818C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60#553—31no—
61D4×Q8—31no—
62#491—31no—
63#485—31no—
64#484—31no—
65#479—31no—
66#468—31no—
67#465—31no—
68#461—31no—
69#448—31no—
70#448D311yescharacteristic, complemented
71#440—31no—
72#435—31no—
73#419—31no—
74C2×#103—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67D411yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
88D3×Q8—22no—
89D3×Q8—41no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C3×SD16—22no—
95C3×SD16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217D411yescharacteristic
104#214—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4D411yescharacteristic
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127C22×Q8—31no—
128C22×Q8C2×D311yescharacteristic
129C2×SD16—31no—
130C2×SD16C2×D311yescharacteristic
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135C4⋊Q8—61no—
136#126—61no—
137#125—31no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#103C2×D311yescharacteristic
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—42no—
177C3×Q8#4112yes—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—42no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—81no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203Dic6—21no—
204Dic6—21no—
205Dic6—42no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—31no—
213C2×Q8—41no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8C22×D311yescharacteristic
218C2×Q8C22×D311yescharacteristic
219C2×D4—21no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C22×C4—21no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229SD16—22no—
230SD16—41no—
231SD16—121no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—31no—
253C22⋊C4—31no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—31no—
260C42—61no—
261C42—61no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3#13712yes—
268C22×C3C22×D411yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—21no—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4C2×#4111yescharacteristic
277C3×C4H(5,𝔽2)11yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—21no—
286C23—31no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—61no—
293Q8—61no—
294Q8—122no—
295Q8#23712yes—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—62no—
300D4—241no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—42no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—122no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—22no—
326C2×C3—81no—
327C2×C3#54611yescharacteristic
328C2×C3C2×#13711yescharacteristic
329C2×C3#57711yescharacteristic
330D3—41no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22#91512yes—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—61no—
346C4—61no—
347C4—61no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23711yescharacteristic
353C4#100411yescharacteristic
354C3#332011yescharacteristic, complemented, Sylow
355C2—22no—
356C2—81no—
357C2—121no—
358C2#559111yescharacteristic
359C2C2×#91511yescharacteristic
360C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5618C2
hypercenter#448D3
hypocenterC3#3320
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
3. C3×#448C2semidirect
4. C2×#835C2semidirect
5. #5174C2semidirect
6. #5170C2semidirect
7. #5164C2semidirect
8. #5162C2semidirect
9. #5041C2semidirect
10. #4814C2semidirect
11. #4802C2semidirect
12. #4797C2semidirect
13. #4796C2semidirect
14. #4730C2semidirect
15. #4722C2semidirect
70. #448D3semidirect
354. C3#3320semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#56182
C3×#4482
C2×#8352
#51742
#51702
#51642
#51622
#50412
#48142
#48022
#47972
#47962
#47302
#47222
#33203