Finite Group Databasebeta

← #74404

Subgroups — #74404

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×#896C211yesmaximal, characteristic, complemented
3C2×#893C211yesmaximal, characteristic, complemented
4C2×#884C211yesmaximal, characteristic, complemented
5C3×#385C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4671C211yesmaximal, characteristic, complemented
13#4667C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2407—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C2×#235—24no—
22C2×#235—24no—
23C2×#235—24no—
24C22×D12C2211yescharacteristic
25C22×D12C2211yescharacteristic
26C22×D12C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×Dic6C2211yescharacteristic
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C4⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×#217C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49#896—24no—
50#893—24no—
51#884—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#386—31no—
70#385—31no—
71#385D311yescharacteristic, complemented
72#380—31no—
73C24×C3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C23×D3C2311yescharacteristic
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—41no—
94C2×#67D414yescomplemented
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100#235—232no—
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—41no—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—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×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126#217—41no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
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×D4—31no—
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⋊C4C2×D311yescharacteristic
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⋊C4—31no—
171C2×C22⋊C4—31no—
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×C3C2×D412yes—
188C23×C3C2411yescharacteristic
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
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×D3C2×D412yes—
208C22×D3C2×D412yes—
209C22×D3#4112yescomplemented
210C2×C3×C4—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×C4—21no—
226C2×C3×C4C2×D412yes—
227C2×C3×C4C2×D412yes—
228C2×C3×C4#4112yescomplemented
229#67—28no—
230#67—28no—
231#67—44no—
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×Dic3C2×D412yes—
244C2×Dic3C2×D412yes—
245C2×Dic3C2×D412yes—
246C2×Dic3C2×D412yes—
247D12—28no—
248D12—28no—
249D12—44no—
250C4×D3—28no—
251C4×D3—28no—
252C4×D3—44no—
253Dic6—28no—
254C24—31no—
255C24—31no—
256C24—31no—
257C24C22×D311yescharacteristic
258#41—632no—
259C2×Q8—34no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×D4—34no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
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—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—34no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
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×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C42—62no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3C22×D411yescharacteristic
329C22×C3C22×D411yescharacteristic
330C22×C3C22×D411yescharacteristic
331C22×C3C2×#4111yescharacteristic
332C22×C3C2×#4111yescharacteristic
333C22×C3H(5,𝔽2)11yescharacteristic
334C22×C32−1+411yescharacteristic
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
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—22no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362Dic3—22no—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3#12014yes—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—32no—
374C23—32no—
375C23—32no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
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—61no—
394C23D3×D412yes—
395C23C23×D311yescharacteristic
396Q8—68no—
397D4—68no—
398D4—68no—
399D4—68no—
400D4—68no—
401D4—124no—
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—21no—
418C2×C4—21no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—32no—
423C2×C4—68no—
424C2×C4—68no—
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—124no—
437C2×C4#23512yescomplemented
438C2×C4D3×D412yes—
439C2×C4D3×D412yes—
440C2×C3—42no—
441C2×C3C2×#12011yescharacteristic
442C2×C3#52511yescharacteristic
443C2×C3#53711yescharacteristic
444C2×C3#54611yescharacteristic
445C2×C3#54711yescharacteristic
446C2×C3#54811yescharacteristic
447C2×C3#55011yescharacteristic
448D3—42no—
449D3—42no—
450D3—42no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—62no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22—121no—
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#100711yescharacteristic
484C22D3×#4111yescharacteristic
485C22#100911yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—34no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—62no—
494C4—121no—
495C4—121no—
496C4—121no—
497C3#240711yescharacteristic, complemented, Sylow
498C2—42no—
499C2—122no—
500C2—122no—
501C2—122no—
502C2#550711yescharacteristic
503C2#554711yescharacteristic
504C2#555311yescharacteristic
505C2D3×#12011yescharacteristic
506C2#562911yescharacteristic
507C2#563311yescharacteristic
508C2#565411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
hypercenter#385D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×#896C2semidirect
3. C2×#893C2semidirect
4. C2×#884C2semidirect
5. C3×#385C2semidirect
6. #5213C2semidirect
7. #4991C2semidirect
8. #4989C2semidirect
9. #4960C2semidirect
10. #4959C2semidirect
11. #4957C2semidirect
12. #4671C2semidirect
13. #4667C2semidirect
14. #4665C2semidirect
15. #4646C2semidirect
71. #385D3semidirect
94. C2×#67D4semidirect
118. C2×C4×D3D4semidirect
121. C2×Dic6D4semidirect
209. C22×D3#41semidirect
228. C2×C3×C4#41semidirect
437. C2×C4#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×#8962
C2×#8932
C2×#8842
C3×#3852
#52132
#49912
#49892
#49602
#49592
#49572
#46712
#46672
#46652
#46462
#24073