Finite Group Databasebeta

← #75856

Subgroups — #75856

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#896C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic, complemented
3C2×#878C211yesmaximal, characteristic, complemented
4C2×C4×D12C211yesmaximal, characteristic, complemented
5C3×#388C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic
8#4959C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4933C211yesmaximal, characteristic, complemented
12#4673C211yesmaximal, characteristic, complemented
13#4672C211yesmaximal, characteristic, complemented
14#4669C211yesmaximal, characteristic, complemented
15#4654C211yesmaximal, characteristic, complemented
16#2463—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C42C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44#896—24no—
45#67⋊C4—24no—
46#878—24no—
47C4×D12—24no—
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C22⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60#394—31no—
61#394—31no—
62#393—31no—
63#392—31no—
64#388—31no—
65#388D311yescharacteristic, complemented
66#387—31no—
67#387—31no—
68#386—31no—
69#386—31no—
70#385—31no—
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—22no—
81C2×#67—22no—
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—41no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C2×D12—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3—22no—
102C2×C4×D3—41no—
103C2×C4×D3—41no—
104C2×Dic6—22no—
105C2×Dic6—22no—
106C2×Dic6—41no—
107C3×C4⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C42—22no—
112#217—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—41no—
121C4⋊Dic3D414yescomplemented
122Dic3⋊C4—22no—
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125Dic3⋊C4D414yescomplemented
126C4×Dic3—22no—
127C22×Q8—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C23×C4—31no—
132C23×C4—31no—
133C23×C4—31no—
134C22⋊Q8—64no—
135C22⋊Q8—64no—
136C4×D4—64no—
137C4×D4—64no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4C2×D311yescharacteristic
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C42—31no—
157C2×C42C2×D311yescharacteristic
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95—31no—
164#95C2×D311yescharacteristic
165#95C2×D311yescharacteristic
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3#4112yes—
173C23×C3C2411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3#4112yes—
187C22×D3#4112yes—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
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—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4C2×D412yes—
205C2×C3×C4C2×D412yes—
206#67—28no—
207#67—44no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—44no—
223C2×Dic3C2×D412yes—
224C2×Dic3C2×D412yes—
225C2×Dic3C2×D412yes—
226C2×Dic3C2×D412yes—
227D12—28no—
228C4×D3—44no—
229C4×D3—44no—
230Dic6—44no—
231C24—31no—
232C24—31no—
233C24C22×D311yescharacteristic
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—121no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—121no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D311yescharacteristic
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D311yescharacteristic
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—34no—
269C4⋊C4—34no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—121no—
285C42—22no—
286C42—62no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3C22×D411yescharacteristic
295C22×C3C22×D411yescharacteristic
296C22×C3C2×#4111yescharacteristic
297C22×C3C2×#4111yescharacteristic
298C22×C3C2×#4111yescharacteristic
299C22×C3H(5,𝔽2)11yescharacteristic
300C22×C32−1+411yescharacteristic
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—22no—
305C2×D3—22no—
306C2×D3—22no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—41no—
321Dic3—22no—
322Dic3—22no—
323Dic3—22no—
324Dic3—22no—
325Dic3—41no—
326Dic3—41no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—32no—
334C23—32no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—61no—
347C23#23512yes—
348C23C23×D311yescharacteristic
349Q8—124no—
350D4—68no—
351D4—68no—
352D4—124no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—124no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4D3×D412yes—
391C2×C4D3×D412yes—
392C2×C3—42no—
393C2×C3#52511yescharacteristic
394C2×C3#52511yescharacteristic
395C2×C3#53911yescharacteristic
396C2×C3D4211yescharacteristic
397C2×C3#54811yescharacteristic
398C2×C3#54811yescharacteristic
399C2×C3#55111yescharacteristic
400D3—42no—
401D3—42no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22—62no—
414C22—62no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22C2×#23511yescharacteristic
424C22C2×D3×D411yescharacteristic
425C22C2×D3×D411yescharacteristic
426C22D3×#4111yescharacteristic
427C22D3×#4111yescharacteristic
428C22#100911yescharacteristic
429C22#101011yescharacteristic
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—62no—
437C4—62no—
438C4—62no—
439C4—62no—
440C4—121no—
441C4—121no—
442C3#246311yescharacteristic, complemented, Sylow
443C2—42no—
444C2—122no—
445C2—122no—
446C2#552711yescharacteristic
447C2#554711yescharacteristic
448C2#555211yescharacteristic
449C2#560911yescharacteristic
450C2#565611yescharacteristic
451C2#567711yescharacteristic
452C2#570711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#388C2
prime radical#4960C2
hypercenter#388D3
hypocenterC3#2463
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#67⋊C4C2semidirect
3. C2×#878C2semidirect
4. C2×C4×D12C2semidirect
5. C3×#388C2semidirect
6. #4989C2semidirect
8. #4959C2semidirect
9. #4958C2semidirect
10. #4956C2semidirect
11. #4933C2semidirect
12. #4673C2semidirect
13. #4672C2semidirect
14. #4669C2semidirect
15. #4654C2semidirect
65. #388D3semidirect
121. C4⋊Dic3D4semidirect
125. Dic3⋊C4D4semidirect
442. C3#2463semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#67⋊C42
C2×#8782
C2×C4×D122
C3×#3882
#49892
#49602
#49592
#49582
#49562
#49332
#46732
#46722
#46692
#46542
#24633