Finite Group Databasebeta

← #80818

Subgroups — #80818

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5629C211yesmaximal, characteristic, complemented
2#5608C211yesmaximal, characteristic, complemented
3C3×#468C211yesmaximal, characteristic, complemented
4#5173C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic, complemented
7#5129C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4884C211yesmaximal, characteristic, complemented
11#4882C211yesmaximal, characteristic, complemented
12#4811C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4781C211yesmaximal, characteristic, complemented
15#4774C211yesmaximal, characteristic, complemented
16#3203—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic
26#939—21no—
27#938—21no—
28#938C2211yescharacteristic
29C2×#213C2211yescharacteristic
30#930C2211yescharacteristic
31#924C2211yescharacteristic
32C22×C3⋊C8C2211yescharacteristic
33C2×#204C2211yescharacteristic
34#897C2211yescharacteristic
35#896—21no—
36#895—21no—
37#895C2211yescharacteristic
38D3×C4⋊C4C2211yescharacteristic
39D3×C4⋊C4C2211yescharacteristic
40#884—21no—
41#884—21no—
42#884—21no—
43#878—21no—
44C3×#107C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#816C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#537—31no—
61#537—31no—
62#484—31no—
63#484—31no—
64#480—31no—
65#480—31no—
66#468—31no—
67#468D311yescharacteristic, complemented
68#467—31no—
69#466—31no—
70#452—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#408—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67D411yescharacteristic
83C2×#67D411yescharacteristic
84C22×Dic3—21no—
85#239—41no—
86#239—41no—
87#237—41no—
88#237—41no—
89#235—41no—
90#235—41no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12D411yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×Dic6D411yescharacteristic
101C2×Dic6C2311yescharacteristic
102C3×SD16—41no—
103C3×M16—41no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#216—41no—
114#213—41no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129#204—41no—
130C2×#41—31no—
131C2×#41—31no—
132C2×#41—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16C2×D311yescharacteristic
136C2×D8—31no—
137C2×M16C2×D311yescharacteristic
138C22×C8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8C2×D311yescharacteristic
145#121—31no—
146#121—31no—
147#121—31no—
148#121—61no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121—61no—
153#121—61no—
154#121C2×D311yescharacteristic
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#107—31no—
160#107C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3C2×D411yescharacteristic
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—81no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—41no—
191#67—41no—
192#67—42no—
193#67—81no—
194#67—81no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201D12—21no—
202D12—21no—
203D12—41no—
204D12—42no—
205D12—81no—
206C4×D3—21no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—42no—
211C4×D3—81no—
212C4×D3—81no—
213Dic6—21no—
214Dic6—21no—
215Dic6—41no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220#41—121no—
221#41—121no—
222#41—121no—
223#41—121no—
224#41—121no—
225#41—121no—
226C2×Q8—31no—
227C2×Q8—31no—
228C2×Q8C22×D311yescharacteristic
229C2×D4—21no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4C22×D311yescharacteristic
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4C22×D311yescharacteristic
247Q16—121no—
248SD16—41no—
249SD16—121no—
250D8—121no—
251M16—41no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—121no—
255C2×C8C22×D311yescharacteristic
256C2×C8C22×D311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—31no—
259C4⋊C4—31no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4C22×D311yescharacteristic
266C4⋊C4C22×D311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3C22×D411yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297Dic3—21no—
298Dic3—21no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—31no—
304C23—31no—
305C23—61no—
306C23D3×D411yescharacteristic
307Q8—21no—
308Q8—21no—
309Q8—61no—
310Q8—61no—
311Q8—121no—
312D4—21no—
313D4—21no—
314D4—61no—
315D4—61no—
316D4—81no—
317D4—121no—
318D4—122no—
319D4—121no—
320D4—121no—
321D4—122no—
322D4—241no—
323D4—241no—
324D4—241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4D3×D411yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—21no—
346C8—21no—
347C8—61no—
348C8—61no—
349C2×C3—41no—
350C2×C3—81no—
351C2×C3#53411yescharacteristic
352C2×C3#57611yescharacteristic
353C2×C3#57811yescharacteristic
354D3—41no—
355D3—41no—
356D3—81no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—61no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100911yescharacteristic
384C3#320311yescharacteristic, complemented, Sylow
385C2—41no—
386C2—81no—
387C2—121no—
388C2—121no—
389C2—241no—
390C2#560511yescharacteristic
391C2#577411yescharacteristic
392C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
hypercenter#468D3
hypocenterC3#3203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #5629C2semidirect
2. #5608C2semidirect
3. C3×#468C2semidirect
4. #5173C2semidirect
5. #5172C2semidirect
6. #5134C2semidirect
7. #5129C2semidirect
8. #5051C2semidirect
9. #5038C2semidirect
10. #4884C2semidirect
11. #4882C2semidirect
12. #4811C2semidirect
13. #4808C2semidirect
14. #4781C2semidirect
15. #4774C2semidirect
67. #468D3semidirect
384. C3#3203semidirect

Maximal subgroups

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

subgroupindex
#56292
#56082
C3×#4682
#51732
#51722
#51342
#51292
#50512
#50382
#48842
#48822
#48112
#48082
#47812
#47742
#32033