Finite Group Databasebeta

← #80066

Subgroups — #80066

Order 384.

482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2C2×#942C211yesmaximal, characteristic, complemented
3C2×#915C211yesmaximal, characteristic, complemented
4#5599C211yesmaximal, characteristic
5C3×#448C211yesmaximal, characteristic, complemented
6#5175C211yesmaximal, characteristic, complemented
7#5173C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4790C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3278—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#237C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946—21no—
25#946C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic
27#944—21no—
28#944C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#216C2211yescharacteristic
31#942—24no—
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic
34#940—21no—
35#940C2211yescharacteristic
36#939—21no—
37C2×#214C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40#924—21no—
41#924C2211yescharacteristic
42C2×#208C2211yescharacteristic
43#915—24no—
44C2×Dic12C2211yescharacteristic
45C2×#204C2211yescharacteristic
46C2×ΓL(1,𝔽25)C2211yescharacteristic
47#895C2211yescharacteristic
48#887C2211yescharacteristic
49#884—21no—
50#67⋊C4—21no—
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic
58Dic3⋊C8C2211yescharacteristic
59C8×Dic3C2211yescharacteristic
60#810C2211yescharacteristic
61#807C2211yescharacteristic
62C2×#137—31no—
63C2×#137—31no—
64#550—31no—
65D4×Q8—31no—
66#492—31no—
67#468—31no—
68#468—31no—
69#462—31no—
70#460—31no—
71#452—31no—
72#452—31no—
73#448—31no—
74#448D311yescharacteristic, complemented
75C4×Q16—31no—
76#436—31no—
77C2×C3×Q8—21no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4C2311yescharacteristic
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67D411yescharacteristic
91C22×Dic3—21no—
92D3×Q8—22no—
93D3×Q8—41no—
94#237—22no—
95#237—41no—
96#235—22no—
97#235—41no—
98C2×D12—21no—
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C2×Dic6—21no—
104C2×Dic6C2311yescharacteristic
105C2×Dic6C2311yescharacteristic
106C3×SD16—22no—
107C3×SD16—41no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112#217—21no—
113#217D411yescharacteristic
114#216—22no—
115#216D414yescomplemented
116#215—22no—
117#215—22no—
118#214—22no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4D411yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3C2311yescharacteristic
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
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143#137—64no—
144#137—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16C2×D311yescharacteristic
153C2×SD16C2×D311yescharacteristic
154C2×M16—31no—
155C2×M16—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#127—31no—
159#127—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163C22⋊Q8—61no—
164#121—31no—
165#121—61no—
166#121—61no—
167#121C2×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#103C2×D311yescharacteristic
178#103C2×D311yescharacteristic
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183C4×C8—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—42no—
192C3×Q8C2×D412yes—
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—81no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—22no—
205#67—42no—
206#67—42no—
207#67—42no—
208#67—81no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—42no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216D12—21no—
217D12—21no—
218D12—81no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—42no—
222C4×D3—42no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—42no—
227Dic6C2×D412yes—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8C2×D412yes—
232#41—62no—
233#41—62no—
234#41—121no—
235#41—121no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—31no—
239C2×Q8—31no—
240C2×Q8—41no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—121no—
244C2×Q8C22×D311yescharacteristic
245C2×Q8C22×D311yescharacteristic
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4C22×D311yescharacteristic
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—34no—
262Q16—62no—
263Q16—62no—
264SD16—22no—
265SD16—41no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270M16—62no—
271M16—62no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4C22×D311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—31no—
287C22⋊C4—31no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—31no—
292C42—61no—
293C42—61no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C22×D411yescharacteristic
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4C22×D411yescharacteristic
312C3×C4H(5,𝔽2)11yescharacteristic
313Dic3—22no—
314Dic3—21no—
315Dic3—21no—
316Dic3—41no—
317Dic3—41no—
318Dic3#13712yescomplemented
319C23—21no—
320C23—31no—
321C23—61no—
322C23D3×D411yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—21no—
326Q8—32no—
327Q8—42no—
328Q8—61no—
329Q8—61no—
330Q8—61no—
331Q8—122no—
332Q8D3×D412yes—
333D4—21no—
334D4—21no—
335D4—62no—
336D4—61no—
337D4—61no—
338D4—81no—
339D4—122no—
340D4—122no—
341D4—122no—
342D4—241no—
343D4—241no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—42no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4D3×D411yescharacteristic
362C2×C4C23×D311yescharacteristic
363C8—21no—
364C8—21no—
365C8—32no—
366C8—61no—
367C2×C3—41no—
368C2×C3—81no—
369C2×C3D4211yescharacteristic
370C2×C3C2×#13711yescharacteristic
371C2×C3#57711yescharacteristic
372D3—41no—
373D3—81no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—22no—
389C4—32no—
390C4—41no—
391C4—41no—
392C4—61no—
393C4—61no—
394C4—62no—
395C4—121no—
396C4—121no—
397C4C2×D3×D411yescharacteristic
398C4#100411yescharacteristic
399C3#327811yescharacteristic, complemented, Sylow
400C2—41no—
401C2—81no—
402C2—121no—
403C2—241no—
404C2#559011yescharacteristic
405C2#576211yescharacteristic
406C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5599C2
hypercenter#448D3
hypocenterC3#3278
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
2. C2×#942C2semidirect
3. C2×#915C2semidirect
5. C3×#448C2semidirect
6. #5175C2semidirect
7. #5173C2semidirect
8. #5163C2semidirect
9. #5042C2semidirect
10. #4812C2semidirect
11. #4806C2semidirect
12. #4795C2semidirect
13. #4790C2semidirect
14. #4733C2semidirect
15. #4729C2semidirect
74. #448D3semidirect
115. #216D4semidirect
318. Dic3#137semidirect
399. C3#3278semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
C2×#9422
C2×#9152
#55992
C3×#4482
#51752
#51732
#51632
#50422
#48122
#48062
#47952
#47902
#47332
#47292
#32783