Finite Group Databasebeta

← #77286

Subgroups — #77286

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#896C211yesmaximal, characteristic, complemented
2C2×#887C211yesmaximal, characteristic, complemented
3C2×#885C211yesmaximal, characteristic, complemented
4C2×#884C211yesmaximal, characteristic
5C3×#396C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4990C211yesmaximal, characteristic, complemented
8#4988C211yesmaximal, characteristic, complemented
9#4976C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4957C211yesmaximal, characteristic, complemented
13#4944C211yesmaximal, characteristic, complemented
14#4676C211yesmaximal, characteristic, complemented
15#4666C211yesmaximal, characteristic, complemented
16#2426—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47C2×C4×Dic3C2211yescharacteristic
48#896—24no—
49#887—24no—
50#885—24no—
51#884—24no—
52C3×#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#124—31no—
57C2×C22⋊Q8—31no—
58C2×#121—31no—
59C2×C4×Q8—31no—
60#396—31no—
61#396D311yescharacteristic, complemented
62#394—31no—
63#394—31no—
64#390—31no—
65#390—31no—
66#390—31no—
67#389—31no—
68#388—31no—
69#388—31no—
70C4×C22⋊C4—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×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—41no—
95C2×C4×D3—22no—
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×Dic6—22no—
99C2×Dic6—22no—
100C2×Dic6D414yescomplemented
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—41no—
107#217—22no—
108#217—22no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—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—41no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121Dic3⋊C4—41no—
122Dic3⋊C4D414yes—
123C4×Dic3—22no—
124C4×Dic3—22no—
125C4×Dic3—22no—
126C4×Dic3—22no—
127C22×Q8—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C23×C4—31no—
132#124—64no—
133C22⋊Q8—64no—
134#121—64no—
135C4×Q8—64no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4C2×D311yescharacteristic
142C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C22⋊C4C2×D311yescharacteristic
158C2×C42—31no—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—31no—
162#95—31no—
163#95—31no—
164#95—31no—
165#95C2×D311yescharacteristic
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
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—21no—
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—44no—
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×Dic3C2×D412yes—
221C2×Dic3C2×D412yes—
222C2×Dic3C2×D412yes—
223C2×Dic3C2×D412yes—
224C2×Dic3#4112yes—
225C2×Dic3#4112yes—
226D12—44no—
227C4×D3—44no—
228Dic6—28no—
229C24—31no—
230C24—31no—
231C24C22×D311yescharacteristic
232C2×Q8—34no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—121no—
243C2×D4—121no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—34no—
262C4⋊C4—41no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—121no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—41no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—121no—
281C42—62no—
282C42—62no—
283C42—62no—
284C42—62no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3C22×D411yescharacteristic
293C22×C3C22×D411yescharacteristic
294C22×C3C2×#4111yescharacteristic
295C22×C3C2×#4111yescharacteristic
296C22×C3C2×#4111yescharacteristic
297C22×C3H(5,𝔽2)11yescharacteristic
298C22×C32−1+411yescharacteristic
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—22no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319Dic3—22no—
320Dic3—22no—
321Dic3—22no—
322Dic3—41no—
323Dic3—41no—
324Dic3#12114yes—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—32no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23C23×D311yescharacteristic
347Q8—68no—
348D4—124no—
349D4—124no—
350D4—124no—
351C2×C4—21no—
352C2×C4—21no—
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—32no—
368C2×C4—32no—
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—124no—
386C2×C4D3×D412yes—
387C2×C4D3×D412yes—
388C2×C3—81no—
389C2×C3C2×#12111yescharacteristic
390C2×C3#52911yescharacteristic
391C2×C3#54211yescharacteristic
392C2×C3#54611yescharacteristic
393C2×C3#54811yescharacteristic
394C2×C3#54811yescharacteristic
395C2×C3#55011yescharacteristic
396D3—42no—
397D3—81no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22C2×D3×D411yescharacteristic
420C22C2×D3×D411yescharacteristic
421C22#100411yescharacteristic
422C22#100711yescharacteristic
423C22D3×#4111yescharacteristic
424C22D3×#4111yescharacteristic
425C22D3×#4111yescharacteristic
426C4—22no—
427C4—22no—
428C4—34no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—62no—
434C4—62no—
435C4—62no—
436C4—121no—
437C4—121no—
438C3#242611yescharacteristic, complemented, Sylow
439C2—81no—
440C2—122no—
441C2—241no—
442C2#559911yescharacteristic
443C2D3×#12111yescharacteristic
444C2#561011yescharacteristic
445C2#563311yescharacteristic
446C2#563311yescharacteristic
447C2#564011yescharacteristic
448C2#570611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
prime radicalC2×#884C2
hypercenter#396D3
hypocenterC3#2426
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#887C2semidirect
3. C2×#885C2semidirect
5. C3×#396C2semidirect
6. #4991C2semidirect
7. #4990C2semidirect
8. #4988C2semidirect
9. #4976C2semidirect
10. #4959C2semidirect
11. #4957C2semidirect
12. #4957C2semidirect
13. #4944C2semidirect
14. #4676C2semidirect
15. #4666C2semidirect
61. #396D3semidirect
100. C2×Dic6D4semidirect
438. C3#2426semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8872
C2×#8852
C2×#8842
C3×#3962
#49912
#49902
#49882
#49762
#49592
#49572
#49442
#46762
#46662
#24263