Finite Group Databasebeta

← #82195

Subgroups — #82195

Order 384.

674 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
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic
3C2×C4×#67C211yesmaximal, characteristic, complemented
4C2×#894C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#5213C211yesmaximal, characteristic, complemented
8#5210C211yesmaximal, characteristic, complemented
9#4993C211yesmaximal, characteristic, complemented
10#4992C211yesmaximal, characteristic, complemented
11#4991C211yesmaximal, characteristic, complemented
12#4990C211yesmaximal, characteristic, complemented
13#4989C211yesmaximal, characteristic, complemented
14#4982C211yesmaximal, characteristic, complemented
15#4980C211yesmaximal, characteristic, complemented
16#2443—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×C4⋊C4—24no—
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31#930—24no—
32C4×#67—24no—
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×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×C4⋊Dic3C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47#894—24no—
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123—31no—
57C2×#121—31no—
58C2×C4×D4—31no—
59C22×C4⋊C4D311yescharacteristic, complemented
60#399—31no—
61#398—31no—
62#397—31no—
63#394—31no—
64#388—31no—
65#387—31no—
66#386—31no—
67#386—31no—
68#386—31no—
69#385—31no—
70#385—31no—
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—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
88C22×C3×C4C2311yescharacteristic
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67D414yes—
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—41no—
104C2×C4×D3—22no—
105C2×C4×D3—22no—
106C2×C4×D3—41no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4D414yescomplemented
111#217—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—22no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—22no—
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125Dic3⋊C4—41no—
126C4×Dic3—22no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C23×C4—31no—
131C23×C4C2×D311yescharacteristic
132C23×C4C2×D311yescharacteristic
133C23×C4C2×D311yescharacteristic
134#123—64no—
135#121—64no—
136C4×D4—64no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4C2×D311yescharacteristic
143C2×C4⋊C4C2×D311yescharacteristic
144C2×C4⋊C4C2×D311yescharacteristic
145C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C42—31no—
158#95—31no—
159#95—31no—
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×C3C2×D412yes—
171C23×C3#4112yes—
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—21no—
187C22×D3C2×D412yes—
188C2×C3×C4—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—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4C2×D412yes—
206C2×C3×C4C2×D412yes—
207C2×C3×C4C2×D412yes—
208C2×C3×C4#4112yes—
209#67—28no—
210#67—44no—
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×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3C2×D412yes—
229D12—44no—
230C4×D3—44no—
231C24—31no—
232C24—31no—
233C24C22×D311yescharacteristic
234C2×D4—34no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—121no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
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×C4—62no—
259C22×C4—62no—
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—22no—
269C4⋊C4—22no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—121no—
274C4⋊C4#6714yescomplemented
275C22⋊C4—62no—
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—62no—
285C22⋊C4—121no—
286C42—62no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3#12314yescomplemented
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—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—
313C2×D3—41no—
314C2×D3—41no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—41no—
320C3×C4—41no—
321Dic3—22no—
322Dic3—41no—
323Dic3—41no—
324Dic3—41no—
325Dic3—41no—
326Dic3—41no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—32no—
332C23—61no—
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#23512yes—
346C23D3×D412yes—
347C23#23912yes—
348C23C23×D311yescharacteristic
349D4—68no—
350D4—124no—
351D4—124no—
352C2×C4—28no—
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—44no—
369C2×C4—44no—
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—61no—
387C2×C4—124no—
388C2×C4#23512yes—
389C2×C4C2×#6712yes—
390C2×C4C2×#6712yes—
391C2×C4C2×#6712yes—
392C2×C3—24no—
393C2×C3C2×#12311yescharacteristic
394C2×C3#52511yescharacteristic
395C2×C3#53911yescharacteristic
396C2×C3D4211yescharacteristic
397C2×C3#54711yescharacteristic
398C2×C3#54811yescharacteristic
399C2×C3#55311yescharacteristic
400D3—42no—
401D3—81no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22#89414yescomplemented
423C22C2×#23511yescharacteristic
424C22C2×#23511yescharacteristic
425C22C2×D3×D411yescharacteristic
426C22#100411yescharacteristic
427C22C2×#23911yescharacteristic
428C22#100711yescharacteristic
429C22C22×#6711yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—62no—
437C4—121no—
438C4—121no—
439C4—121no—
440C4—121no—
441C4—121no—
442C3#244311yescharacteristic, complemented, Sylow
443C2—24no—
444C2—122no—
445C2—241no—
446C2C2×#89411yescharacteristic
447C2#550711yescharacteristic
448C2#551411yescharacteristic
449C2#551511yescharacteristic
450C2#579111yescharacteristic
451C2D4×#6711yescharacteristic
452C2#581711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC2×#930C2
hypercenterC22×C4⋊C4D3
hypocenterC3#2443
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
3. C2×C4×#67C2semidirect
4. C2×#894C2semidirect
5. #5213C2semidirect
6. #5213C2semidirect
7. #5213C2semidirect
8. #5210C2semidirect
9. #4993C2semidirect
10. #4992C2semidirect
11. #4991C2semidirect
12. #4990C2semidirect
13. #4989C2semidirect
14. #4982C2semidirect
15. #4980C2semidirect
59. C22×C4⋊C4D3semidirect
110. C3×C4⋊C4D4semidirect
274. C4⋊C4#67semidirect
293. C22×C3#123semidirect
422. C22#894semidirect
442. C3#2443semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×#9302
C2×C4×#672
C2×#8942
#52132
#52102
#49932
#49922
#49912
#49902
#49892
#49822
#49802
#24433