Finite Group Databasebeta

← #76702

Subgroups — #76702

Order 384.

620 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#939C211yesmaximal, characteristic, complemented
2C2×D4×Dic3C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5220C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4958C211yesmaximal, characteristic, complemented
11#4951C211yesmaximal, characteristic, complemented
12#4949C211yesmaximal, characteristic, complemented
13#4943C211yesmaximal, characteristic, complemented
14#4669C211yesmaximal, characteristic, complemented
15#4660C211yesmaximal, characteristic, complemented
16#2635—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35#939—24no—
36D4×Dic3—24no—
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×C4⋊Dic3C2211yescharacteristic
42C2×C4⋊Dic3C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#121—31no—
55C2×C4×D4—31no—
56#400—31no—
57#399—31no—
58#394—31no—
59#394—31no—
60#394—31no—
61#392—31no—
62#392—31no—
63#392D311yescharacteristic, complemented
64#388—31no—
65#388—31no—
66#386—31no—
67#386—31no—
68#381—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—41no—
90C2×#67—41no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—41no—
95C22×Dic3—22no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3D414yescomplemented
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C3×C22⋊C4—22no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111#217—22no—
112#217—22no—
113#217—22no—
114#217—41no—
115#217—41no—
116#217—41no—
117#217—41no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—22no—
123C4⋊Dic3—41no—
124Dic3⋊C4—22no—
125Dic3⋊C4—41no—
126C4×Dic3—22no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4C2×D311yescharacteristic
131C22×D4C2×D311yescharacteristic
132C22×D4C2×D311yescharacteristic
133C23×C4—31no—
134C23×C4—31no—
135#121—64no—
136C4×D4—64no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C42—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95C2×D311yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3C2×D412yes—
184C23×C3#4112yes—
185C23×C3C2411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C3×D4—28no—
194C3×D4—44no—
195C3×D4—44no—
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—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4#4112yes—
208#67—44no—
209#67—44no—
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×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—44no—
231C2×Dic3—44no—
232C2×Dic3C2×D412yes—
233C2×Dic3C2×D412yes—
234C24—31no—
235C24C22×D311yescharacteristic
236C24C22×D311yescharacteristic
237C24C22×D311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—62no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—34no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—62no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—62no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3C22×D411yescharacteristic
314C22×C3C2×#4111yescharacteristic
315C22×C3C2×#4111yescharacteristic
316C22×C3H(5,𝔽2)11yescharacteristic
317C22×C3H(5,𝔽2)11yescharacteristic
318C22×C3H(5,𝔽2)11yescharacteristic
319C22×C3H(5,𝔽2)11yescharacteristic
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C3×C4—22no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331Dic3—22no—
332Dic3—22no—
333Dic3—41no—
334Dic3—41no—
335Dic3—41no—
336Dic3—41no—
337Dic3—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23D3×D412yes—
365C23#23712yes—
366C23C23×D311yescharacteristic
367D4—28no—
368D4—44no—
369D4—44no—
370D4—124no—
371D4—124no—
372D4—124no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4#23712yes—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—81no—
411C2×C3#52511yescharacteristic
412C2×C3#53411yescharacteristic
413C2×C3#53511yescharacteristic
414C2×C3#53811yescharacteristic
415C2×C3#53811yescharacteristic
416C2×C3#54011yescharacteristic
417C2×C3#54011yescharacteristic
418D3—81no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22C2×D3×D411yescharacteristic
448C22C2×#23711yescharacteristic
449C22C2×#23711yescharacteristic
450C22#100411yescharacteristic
451C22#100411yescharacteristic
452C22#100411yescharacteristic
453C22#100911yescharacteristic
454C4—22no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—62no—
459C4—62no—
460C4—121no—
461C4—121no—
462C4—121no—
463C4—121no—
464C4—121no—
465C3#263511yescharacteristic, complemented, Sylow
466C2—42no—
467C2—42no—
468C2—81no—
469C2—241no—
470C2#558611yescharacteristic
471C2#559211yescharacteristic
472C2#559311yescharacteristic
473C2#560511yescharacteristic
474C2#561711yescharacteristic
475C2#561911yescharacteristic
476C2#571911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2635
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#939C2semidirect
2. C2×D4×Dic3C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #5220C2semidirect
8. #4961C2semidirect
9. #4961C2semidirect
10. #4958C2semidirect
11. #4951C2semidirect
12. #4949C2semidirect
13. #4943C2semidirect
14. #4669C2semidirect
15. #4660C2semidirect
63. #392D3semidirect
100. C22×Dic3D4semidirect
465. C3#2635semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×D4×Dic32
C3×#3922
#52232
#52222
#52202
#49612
#49582
#49512
#49492
#49432
#46692
#46602
#26353