Finite Group Databasebeta

← #76894

Subgroups — #76894

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×#237C211yesmaximal, characteristic, complemented
2C2×#896C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C2×#67⋊C4C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4958C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4946C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2407—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C23×Dic3C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C2×#237—24no—
25C2×#237—24no—
26C2×#237—24no—
27C22×D12C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×Dic6C2211yescharacteristic
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49#896—24no—
50#884—24no—
51#67⋊C4—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#394D311yescharacteristic, complemented
64#392—31no—
65#392—31no—
66#391—31no—
67#390—31no—
68#386—31no—
69#386—31no—
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—22no—
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—22no—
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—41no—
96C2×#67D414yescomplemented
97C22×Dic3—22no—
98C22×Dic3—22no—
99C22×Dic3—22no—
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3D414yescomplemented
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109#237—232no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—41no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6D414yescomplemented
122C3×C4⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127#217—22no—
128#217—41no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—41no—
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×C4—31no—
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⋊C4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
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#95—31no—
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×C3C2×D412yes—
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×D3#4112yescomplemented
210C3×D4—28no—
211C2×C3×C4—21no—
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—
226#67—28no—
227#67—28no—
228#67—28no—
229#67—44no—
230C2×Dic3—28no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—28no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3C2×D412yes—
245C2×Dic3C2×D412yes—
246C2×Dic3C2×D412yes—
247C2×Dic3C2×D412yes—
248C2×Dic3C2×D412yes—
249C2×Dic3#4112yescomplemented
250D12—44no—
251C4×D3—28no—
252C4×D3—44no—
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—22no—
265C2×D4—34no—
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—121no—
279C2×D4—121no—
280C22×C4—31no—
281C22×C4—34no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
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—22no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
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—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361Dic3—22no—
362Dic3—22no—
363Dic3—22no—
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—
393C23D3×D412yes—
394C23D3×D412yes—
395C23C23×D311yescharacteristic
396Q8—68no—
397D4—28no—
398D4—68no—
399D4—68no—
400D4—68no—
401D4—124no—
402D4—124no—
403C2×C4—21no—
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—32no—
423C2×C4—68no—
424C2×C4—68no—
425C2×C4—68no—
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—61no—
438C2×C4—124no—
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×D3×D411yescharacteristic
480C22C2×D3×D411yescharacteristic
481C22C2×D3×D411yescharacteristic
482C22#100411yescharacteristic
483C22D3×#4111yescharacteristic
484C22D3×#4111yescharacteristic
485C22#101011yescharacteristic
486C4—22no—
487C4—34no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—62no—
493C4—62no—
494C4—62no—
495C4—121no—
496C4—121no—
497C3#240711yescharacteristic, complemented, Sylow
498C2—42no—
499C2—42no—
500C2—122no—
501C2—122no—
502C2D3×#12011yescharacteristic
503C2#559711yescharacteristic
504C2#559911yescharacteristic
505C2#561011yescharacteristic
506C2#565511yescharacteristic
507C2#565611yescharacteristic
508C2#567711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#896C2semidirect
3. C2×#884C2semidirect
4. C2×#67⋊C4C2semidirect
5. C3×#394C2semidirect
6. #5223C2semidirect
7. #4989C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4958C2semidirect
11. #4957C2semidirect
12. #4946C2semidirect
13. #4942C2semidirect
14. #4665C2semidirect
15. #4646C2semidirect
63. #394D3semidirect
96. C2×#67D4semidirect
102. C22×Dic3D4semidirect
121. C2×Dic6D4semidirect
209. C22×D3#41semidirect
249. C2×Dic3#41semidirect
497. C3#2407semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#8962
C2×#8842
C2×#67⋊C42
C3×#3942
#52232
#49892
#49602
#49592
#49582
#49572
#49462
#49422
#46652
#46462
#24073