Finite Group Databasebeta

← #74688

Subgroups — #74688

Order 384.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4672C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4667C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2407—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C2×#235—24no—
25C2×#235—24no—
26C2×#235—24no—
27C22×D12C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×Dic6C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37#939—24no—
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×C4⋊Dic3C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50#67⋊C4—24no—
51#878—24no—
52C3×#95C2211yescharacteristic
53C3×#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C3⋊#95C2211yescharacteristic
58C22×#41—31no—
59C2×C22⋊Q8—31no—
60C2×#121—31no—
61C2×C4×D4—31no—
62#394—31no—
63#394—31no—
64#392—31no—
65#392—31no—
66#391—31no—
67#390—31no—
68#386—31no—
69#386D311yescharacteristic, complemented
70#385—31no—
71#385—31no—
72#380—31no—
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C23×D3C2311yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—41no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
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—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—41no—
99C2×#67D414yescomplemented
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3—22no—
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109#235—232no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3D414yescomplemented
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6D414yescomplemented
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—41no—
125#217—22no—
126#217—22no—
127#217—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134C4⋊Dic3—22no—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139C4×Dic3—22no—
140C2×#41—64no—
141C2×#41—64no—
142C2×#41—64no—
143C22×Q8—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4C2×D311yescharacteristic
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4C2×D311yescharacteristic
154C22⋊Q8—64no—
155#121—64no—
156C4×D4—64no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4C2×D311yescharacteristic
172C2×C22⋊C4C2×D311yescharacteristic
173C2×C22⋊C4C2×D311yescharacteristic
174C2×C42—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178#95—31no—
179#95C2×D311yescharacteristic
180#95C2×D311yescharacteristic
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3C2×D412yes—
194C23×C3#4112yescomplemented
195C23×C3C2411yescharacteristic
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3C2×D412yes—
209C22×D3C2×D412yes—
210C3×D4—44no—
211C2×C3×C4—28no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4C2×D412yes—
226C2×C3×C4C2×D412yes—
227#67—28no—
228#67—28no—
229#67—28no—
230#67—44no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—44no—
245C2×Dic3C2×D412yes—
246C2×Dic3C2×D412yes—
247C2×Dic3C2×D412yes—
248C2×Dic3C2×D412yes—
249C2×Dic3#4112yescomplemented
250D12—28no—
251C4×D3—28no—
252C4×D3—28no—
253Dic6—28no—
254C24—31no—
255C24—31no—
256C24C22×D311yescharacteristic
257C24C22×D311yescharacteristic
258#41—632no—
259C2×Q8—34no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—34no—
265C2×D4—41no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—121no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—34no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C42—62no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3C22×D411yescharacteristic
336C22×C3C22×D411yescharacteristic
337C22×C3C22×D411yescharacteristic
338C22×C3C2×#4111yescharacteristic
339C22×C3C2×#4111yescharacteristic
340C22×C3H(5,𝔽2)11yescharacteristic
341C22×C32−1+411yescharacteristic
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—22no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361Dic3—22no—
362Dic3—22no—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3#12014yes—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—32no—
380C23—32no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23#23512yescomplemented
394C23D3×D412yes—
395C23C23×D311yescharacteristic
396Q8—68no—
397D4—44no—
398D4—68no—
399D4—68no—
400D4—68no—
401D4—68no—
402D4—124no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—68no—
423C2×C4—68no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—124no—
438C2×C4D3×D412yes—
439C2×C4D3×D412yes—
440C2×C3—42no—
441C2×C3—42no—
442C2×C3C2×#12011yescharacteristic
443C2×C3#52511yescharacteristic
444C2×C3#53711yescharacteristic
445C2×C3#54611yescharacteristic
446C2×C3#54711yescharacteristic
447C2×C3#54811yescharacteristic
448C2×C3#55011yescharacteristic
449D3—42no—
450D3—42no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—62no—
466C22—62no—
467C22—62no—
468C22—62no—
469C22—62no—
470C22—62no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22C2×#23511yescharacteristic
480C22C2×D3×D411yescharacteristic
481C22C2×D3×D411yescharacteristic
482C22C2×D3×D411yescharacteristic
483C22#100411yescharacteristic
484C22D3×#4111yescharacteristic
485C22#101011yescharacteristic
486C4—22no—
487C4—22no—
488C4—34no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—62no—
493C4—62no—
494C4—121no—
495C4—121no—
496C4—121no—
497C3#240711yescharacteristic, complemented, Sylow
498C2—42no—
499C2—42no—
500C2—122no—
501C2—122no—
502C2#549011yescharacteristic
503C2#554711yescharacteristic
504C2#555211yescharacteristic
505C2D3×#12011yescharacteristic
506C2#559911yescharacteristic
507C2#560811yescharacteristic
508C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
hypercenter#386D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×#939C2semidirect
3. C2×#67⋊C4C2semidirect
4. C2×#878C2semidirect
5. C3×#386C2semidirect
6. #5223C2semidirect
7. #5210C2semidirect
8. #4961C2semidirect
9. #4960C2semidirect
10. #4957C2semidirect
11. #4956C2semidirect
12. #4672C2semidirect
13. #4668C2semidirect
14. #4667C2semidirect
15. #4646C2semidirect
69. #386D3semidirect
99. C2×#67D4semidirect
118. C2×C4×D3D4semidirect
121. C2×Dic6D4semidirect
194. C23×C3#41semidirect
249. C2×Dic3#41semidirect
393. C23#235semidirect
497. C3#2407semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#9392
C2×#67⋊C42
C2×#8782
C3×#3862
#52232
#52102
#49612
#49602
#49572
#49562
#46722
#46682
#46672
#46462
#24073