Finite Group Databasebeta

← #82367

Subgroups — #82367

Order 384.

646 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic, complemented
4C2×#885C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic
6#5222C211yesmaximal, characteristic, complemented
7#5213C211yesmaximal, characteristic, complemented
8#5210C211yesmaximal, characteristic, complemented
9#4994C211yesmaximal, characteristic, complemented
10#4979C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4951C211yesmaximal, characteristic, complemented
13#4944C211yesmaximal, characteristic, complemented
14#4940C211yesmaximal, characteristic, complemented
15C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
16#2461—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C3×C4×D4—24no—
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C42C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34#929—24no—
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41#924—24no—
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×C4×Dic3C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49#885—24no—
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#124—31no—
57C2×#123—31no—
58C2×C22⋊Q8—31no—
59C2×C4×D4D311yescharacteristic, complemented
60#396—31no—
61#395—31no—
62#394—31no—
63#392—31no—
64#390—31no—
65#388—31no—
66#388—31no—
67#388—31no—
68#386—31no—
69#385—31no—
70C4×C4⋊C4—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—41no—
82C22×C3×C4—41no—
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—41no—
93C2×#67—41no—
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6—41no—
104C3×C4⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4D414yescomplemented
107C3×C42—22no—
108#217—22no—
109#217—22no—
110#217—22no—
111#217—22no—
112#217—22no—
113#217—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—
119C4⋊Dic3—22no—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122Dic3⋊C4—22no—
123Dic3⋊C4—41no—
124Dic3⋊C4D414yescomplemented
125C4×Dic3—22no—
126C4×Dic3—22no—
127C22×Q8—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4C2×D311yescharacteristic
131C23×C4C2×D311yescharacteristic
132C23×C4C2×D311yescharacteristic
133#124—64no—
134#123—64no—
135C22⋊Q8—64no—
136C4×D4—24no—
137C2×C4⋊C4—31no—
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⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C42—31no—
158C2×C42—31no—
159C2×C42C2×D311yescharacteristic
160#95—31no—
161#95—31no—
162#95—31no—
163#95—31no—
164#95—31no—
165#95—31no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3C2×D412yes—
179C23×C3#4112yes—
180C23×C3C2411yescharacteristic
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C3×D4—28no—
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—44no—
201C2×C3×C4—44no—
202C2×C3×C4C2×D412yes—
203C2×C3×C4C2×D412yes—
204C2×C3×C4C2×D412yes—
205C2×C3×C4#4112yes—
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×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3C2×D412yes—
227C2×Dic3C2×D412yes—
228C2×Dic3#4112yes—
229Dic6—44no—
230C24—31no—
231C24C22×D311yescharacteristic
232C24C22×D311yescharacteristic
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8—121no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—121no—
244C2×D4—121no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22×D311yescharacteristic
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D311yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—34no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—121no—
270C22⋊C4—22no—
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—62no—
281C22⋊C4—62no—
282C22⋊C4#6714yescomplemented
283C42—22no—
284C42—62no—
285C42—62no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C22×D411yescharacteristic
301C22×C3C22×D411yescharacteristic
302C22×C3C2×#4111yescharacteristic
303C22×C3C2×#4111yescharacteristic
304C22×C3C2×#4111yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C22×C32−1+411yescharacteristic
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—41no—
319Dic3—22no—
320Dic3—22no—
321Dic3—22no—
322Dic3—41no—
323Dic3—41no—
324Dic3—41no—
325Dic3—41no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23#23512yes—
346C23C2×#6712yes—
347C23C23×D311yescharacteristic
348Q8—124no—
349D4—28no—
350D4—124no—
351D4—124no—
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—32no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
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#23512yes—
387C2×C4D3×D412yes—
388C2×C4C2×#6712yes—
389C2×C4C2×#6712yes—
390C2×C3—42no—
391C2×C3—42no—
392C2×C3#52511yescharacteristic
393C2×C3#53211yescharacteristic
394C2×C3#54211yescharacteristic
395C2×C3D4211yescharacteristic
396C2×C3#54711yescharacteristic
397C2×C3#54811yescharacteristic
398C2×C3#55111yescharacteristic
399D3—81no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22C2×#23511yescharacteristic
422C22C2×#23511yescharacteristic
423C22C2×D3×D411yescharacteristic
424C22#100411yescharacteristic
425C22D3×#4111yescharacteristic
426C22#101011yescharacteristic
427C22C22×#6711yescharacteristic
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—62no—
434C4—62no—
435C4—62no—
436C4—121no—
437C4—121no—
438C4—121no—
439C4—121no—
440C3#246111yescharacteristic, complemented, Sylow
441C2—42no—
442C2—42no—
443C2—241no—
444C2#554811yescharacteristic
445C2#555211yescharacteristic
446C2#555611yescharacteristic
447C2#556111yescharacteristic
448C2#579111yescharacteristic
449C2D4×#6711yescharacteristic
450C2#582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radical#5223C2
hypercenterC2×C4×D4D3
hypocenterC3#2461
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#929C2semidirect
3. C2×#924C2semidirect
4. C2×#885C2semidirect
6. #5222C2semidirect
7. #5213C2semidirect
8. #5210C2semidirect
9. #4994C2semidirect
10. #4979C2semidirect
11. #4957C2semidirect
12. #4951C2semidirect
13. #4944C2semidirect
14. #4940C2semidirect
15. C4×Dic3⋊C4C2semidirect
59. C2×C4×D4D3semidirect
106. C3×C22⋊C4D4semidirect
124. Dic3⋊C4D4semidirect
282. C22⋊C4#67semidirect
440. C3#2461semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9292
C2×#9242
C2×#8852
#52232
#52222
#52132
#52102
#49942
#49792
#49572
#49512
#49442
#49402
C4×Dic3⋊C42
#24613