Finite Group Databasebeta

← #76668

Subgroups — #76668

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×#939C211yesmaximal, characteristic, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#392C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4950C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2407—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C23×Dic3C2211yescharacteristic
26C2×#237—24no—
27C2×#237—24no—
28C2×#237—24no—
29C22×C4×D3C2211yescharacteristic
30C22×Dic6C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40#939—24no—
41D4×Dic3—24no—
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊Dic3C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×Dic3⋊C4C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
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#392D311yescharacteristic, complemented
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
75C24×C3C2311yescharacteristic
76C23×D3C2311yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—41no—
98C2×#67D414yescomplemented
99C22×Dic3—22no—
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3—22no—
103C22×Dic3—22no—
104C22×Dic3—22no—
105C22×Dic3—22no—
106C22×Dic3—22no—
107C22×Dic3D414yescomplemented
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115#237—232no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
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#217—22no—
129#217—41no—
130#217—41no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—41no—
134C4⋊Dic3—22no—
135C4⋊Dic3—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×D4C2×D311yescharacteristic
148C22×D4C2×D311yescharacteristic
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⋊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#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×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3C2×D412yes—
200C23×C3C2×D412yes—
201C23×C3#4112yescomplemented
202C23×C3C2411yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3C2×D412yes—
210C3×D4—28no—
211C3×D4—28no—
212C3×D4—44no—
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×C4C2×D412yes—
224C2×C3×C4#4112yescomplemented
225#67—28no—
226#67—28no—
227#67—44no—
228C2×Dic3—28no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—28no—
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—21no—
245C2×Dic3—21no—
246C2×Dic3—44no—
247C2×Dic3C2×D412yes—
248C2×Dic3C2×D412yes—
249C2×Dic3C2×D412yes—
250C2×Dic3C2×D412yes—
251C2×Dic3C2×D412yes—
252C4×D3—28no—
253Dic6—28no—
254C24—31no—
255C24C22×D311yescharacteristic
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—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—34no—
271C2×D4—41no—
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—31no—
281C22×C4—34no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
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×C4—62no—
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—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3C22×D411yescharacteristic
343C22×C3C22×D411yescharacteristic
344C22×C3C22×D411yescharacteristic
345C22×C3C2×#4111yescharacteristic
346C22×C3C2×#4111yescharacteristic
347C22×C3H(5,𝔽2)11yescharacteristic
348C22×C32−1+411yescharacteristic
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—41no—
360Dic3—22no—
361Dic3—22no—
362Dic3—41no—
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—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—32no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23D3×D412yes—
393C23D3×D412yes—
394C23#23712yescomplemented
395C23C23×D311yescharacteristic
396Q8—68no—
397D4—28no—
398D4—28no—
399D4—44no—
400D4—68no—
401D4—68no—
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—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—68no—
419C2×C4—68no—
420C2×C4—68no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
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×C4#23712yescomplemented
440C2×C3—42no—
441C2×C3—42no—
442C2×C3—42no—
443C2×C3C2×#12011yescharacteristic
444C2×C3#52511yescharacteristic
445C2×C3#53711yescharacteristic
446C2×C3#54611yescharacteristic
447C2×C3#54711yescharacteristic
448C2×C3#54811yescharacteristic
449C2×C3#55011yescharacteristic
450D3—42no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—62no—
473C22—62no—
474C22—62no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22C2×D3×D411yescharacteristic
480C22C2×D3×D411yescharacteristic
481C22C2×D3×D411yescharacteristic
482C22C2×#23711yescharacteristic
483C22C2×#23711yescharacteristic
484C22#100411yescharacteristic
485C22#101011yescharacteristic
486C4—22no—
487C4—22no—
488C4—34no—
489C4—41no—
490C4—41no—
491C4—62no—
492C4—62no—
493C4—121no—
494C4—121no—
495C4—121no—
496C4—121no—
497C3#240711yescharacteristic, complemented, Sylow
498C2—42no—
499C2—42no—
500C2—42no—
501C2—122no—
502C2#558611yescharacteristic
503C2D3×#12011yescharacteristic
504C2#559111yescharacteristic
505C2#559711yescharacteristic
506C2#559811yescharacteristic
507C2#561111yescharacteristic
508C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#939C2semidirect
3. C2×D4×Dic3C2semidirect
4. C2×#878C2semidirect
5. C3×#392C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5220C2semidirect
9. #4961C2semidirect
10. #4957C2semidirect
11. #4956C2semidirect
12. #4950C2semidirect
13. #4942C2semidirect
14. #4668C2semidirect
15. #4646C2semidirect
65. #392D3semidirect
98. C2×#67D4semidirect
107. C22×Dic3D4semidirect
121. C2×Dic6D4semidirect
201. C23×C3#41semidirect
224. C2×C3×C4#41semidirect
394. C23#237semidirect
439. C2×C4#237semidirect
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×#9392
C2×D4×Dic32
C2×#8782
C3×#3922
#52232
#52222
#52202
#49612
#49572
#49562
#49502
#49422
#46682
#46462
#24073