Finite Group Databasebeta

← #80036

Subgroups — #80036

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5816C211yesmaximal, characteristic, complemented
2#5619C211yesmaximal, characteristic, complemented
3C3×#448C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5176C211yesmaximal, characteristic, complemented
6#5175C211yesmaximal, characteristic, complemented
7#5165C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4818C211yesmaximal, characteristic, complemented
12#4800C211yesmaximal, characteristic, complemented
13#4797C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3221—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#947C2211yescharacteristic
23#947C2211yescharacteristic
24#946—21no—
25#946—21no—
26#944C2211yescharacteristic
27#944C2211yescharacteristic
28C2×#215C2211yescharacteristic
29C2×#215C2211yescharacteristic
30#940C2211yescharacteristic
31#939—21no—
32#938—21no—
33D4×Dic3C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#930C2211yescharacteristic
37#929—21no—
38#929—21no—
39#926C2211yescharacteristic
40C2×D24C2211yescharacteristic
41C2×#204C2211yescharacteristic
42#893C2211yescharacteristic
43#890C2211yescharacteristic
44#886—21no—
45#883—21no—
46C3×#103C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#810C2211yescharacteristic
59#809C2211yescharacteristic
60#540—31no—
61#536—31no—
62#496—31no—
63#492—31no—
64#491—31no—
65#488—31no—
66#463—31no—
67#463—31no—
68#461—31no—
69#461—31no—
70#449—31no—
71#449—31no—
72#448—31no—
73#448D311yescharacteristic, complemented
74#433—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—41no—
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×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C22×Dic3—21no—
89#235—41no—
90#235—41no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×Dic6C2311yescharacteristic
96C3×SD16—41no—
97C3×SD16—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102#217—21no—
103#217D411yescharacteristic
104#217D411yescharacteristic
105#215—41no—
106#215—41no—
107#214—41no—
108#213—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3D411yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129#204—41no—
130C2×#41—31no—
131C22×Q8C2×D311yescharacteristic
132C2×SD16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic
137C2×SD16C2×D311yescharacteristic
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142#127—31no—
143#125—31no—
144#124—31no—
145#124—31no—
146#123—61no—
147#123—61no—
148#123—61no—
149#123—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152#121—31no—
153#121—61no—
154#121—61no—
155#121C2×D311yescharacteristic
156C4×D4—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—42no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—81no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—81no—
195#67—81no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—81no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208C4×D3—42no—
209C4×D3—81no—
210Dic6—21no—
211Dic6—21no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216#41—121no—
217#41—121no—
218C2×Q8—21no—
219C2×Q8—31no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8C22×D311yescharacteristic
223C2×Q8C22×D311yescharacteristic
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4C22×D311yescharacteristic
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C22×D311yescharacteristic
237SD16—41no—
238SD16—41no—
239SD16—121no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243D8—121no—
244D8—121no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8C22×D311yescharacteristic
248C2×C8C22×D311yescharacteristic
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C42—31no—
270C42—31no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4H(5,𝔽2)11yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296C23—21no—
297C23—61no—
298C23—61no—
299C23D3×D411yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—42no—
305Q8—61no—
306Q8—61no—
307D4—21no—
308D4—21no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—61no—
313D4—81no—
314D4—122no—
315D4—241no—
316D4—241no—
317D4—241no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4D3×D411yescharacteristic
334C2×C4C23×D311yescharacteristic
335C8—21no—
336C8—21no—
337C8—61no—
338C8—61no—
339C2×C3—41no—
340C2×C3—81no—
341C2×C3#53511yescharacteristic
342C2×C3#57711yescharacteristic
343C2×C3#57711yescharacteristic
344D3—81no—
345D3—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22C2×D3×D411yescharacteristic
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—61no—
364C4—61no—
365C4—61no—
366C4—61no—
367C4—121no—
368C4—121no—
369C4#100411yescharacteristic
370C4#100411yescharacteristic
371C3#322111yescharacteristic, complemented, Sylow
372C2—41no—
373C2—81no—
374C2—241no—
375C2—241no—
376C2#559211yescharacteristic
377C2#576211yescharacteristic
378C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
hypercenter#448D3
hypocenterC3#3221
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. #5816C2semidirect
2. #5619C2semidirect
3. C3×#448C2semidirect
4. #5227C2semidirect
5. #5176C2semidirect
6. #5175C2semidirect
7. #5165C2semidirect
8. #5162C2semidirect
9. #5037C2semidirect
10. #4819C2semidirect
11. #4818C2semidirect
12. #4800C2semidirect
13. #4797C2semidirect
14. #4734C2semidirect
15. #4720C2semidirect
73. #448D3semidirect
371. C3#3221semidirect

Maximal subgroups

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

subgroupindex
#58162
#56192
C3×#4482
#52272
#51762
#51752
#51652
#51622
#50372
#48192
#48182
#48002
#47972
#47342
#47202
#32213