Finite Group Databasebeta

← #82696

Subgroups — #82696

Order 384.

598 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#898C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5221C211yesmaximal, characteristic, complemented
8#5221C211yesmaximal, characteristic, complemented
9#5213C211yesmaximal, characteristic, complemented
10#5210C211yesmaximal, characteristic, complemented
11#4980C211yesmaximal, characteristic, complemented
12#4961C211yesmaximal, characteristic, complemented
13#4957C211yesmaximal, characteristic, complemented
14#4953C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2714—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C3×#121—24no—
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
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
46C2×C4×Dic3C2211yescharacteristic
47#898—24no—
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#126—31no—
55C2×#121D311yescharacteristic, complemented
56#397—31no—
57#394—31no—
58#394—31no—
59#392—31no—
60#392—31no—
61#392—31no—
62#390—31no—
63#390—31no—
64#390—31no—
65#388—31no—
66#387—31no—
67#386—31no—
68#385—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×C4—22no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C3×C4⋊C4D414yescomplemented
106C3×C22⋊C4—22no—
107C3×C22⋊C4—22no—
108#217—22no—
109#217—22no—
110#217—41no—
111#217—41no—
112#217—41no—
113#217—41no—
114#217—41no—
115#217—41no—
116#217—41no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—41no—
121C4⋊Dic3—22no—
122C4⋊Dic3—41no—
123Dic3⋊C4—22no—
124Dic3⋊C4—41no—
125C4×Dic3—22no—
126C4×Dic3—41no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4C2×D311yescharacteristic
131C22×D4C2×D311yescharacteristic
132C22×D4C2×D311yescharacteristic
133C23×C4C2×D311yescharacteristic
134#126—64no—
135#121—24no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C42—31no—
157C2×C42—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95—31no—
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×C3—21no—
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—44no—
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—44no—
206C2×C3×C4C2×D412yes—
207C2×C3×C4C2×D412yes—
208C2×C3×C4C2×D412yes—
209#67—44no—
210#67—44no—
211#67—44no—
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—21no—
231C2×Dic3—21no—
232C2×Dic3#4112yes—
233C24—31no—
234C24C22×D311yescharacteristic
235C24C22×D311yescharacteristic
236C24C22×D311yescharacteristic
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C22×C4—22no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4#6714yescomplemented
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—62no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C42—62no—
290C42—121no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
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×C3C22×D411yescharacteristic
313C22×C3C2×#4111yescharacteristic
314C22×C3C2×#4111yescharacteristic
315C22×C3H(5,𝔽2)11yescharacteristic
316C22×C3H(5,𝔽2)11yescharacteristic
317C22×C3H(5,𝔽2)11yescharacteristic
318C22×C3H(5,𝔽2)11yescharacteristic
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—41no—
330Dic3—22no—
331Dic3—41no—
332Dic3—41no—
333Dic3—41no—
334Dic3—41no—
335Dic3—41no—
336Dic3—41no—
337C23—21no—
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—
364C23#23712yes—
365C23C23×D311yescharacteristic
366D4—44no—
367D4—44no—
368D4—44no—
369D4—124no—
370D4—124no—
371D4—124no—
372C2×C4—21no—
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—32no—
382C2×C4—44no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
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×C4C2×#6712yes—
404C2×C4C2×#6712yes—
405C2×C4C2×#6712yes—
406C2×C3—42no—
407C2×C3—81no—
408C2×C3—81no—
409C2×C3#53511yescharacteristic
410C2×C3#53811yescharacteristic
411C2×C3#53811yescharacteristic
412C2×C3#54611yescharacteristic
413C2×C3#55011yescharacteristic
414C2×C3#55311yescharacteristic
415C2×C3#56011yescharacteristic
416D3—81no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
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—121no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22C2×#23711yescharacteristic
446C22#100411yescharacteristic
447C22#100411yescharacteristic
448C22#100411yescharacteristic
449C22D3×#4111yescharacteristic
450C22#100911yescharacteristic
451C22C22×#6711yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—62no—
457C4—121no—
458C4—121no—
459C4—121no—
460C4—121no—
461C4—121no—
462C4—121no—
463C3#271411yescharacteristic, complemented, Sylow
464C2—42no—
465C2—81no—
466C2—81no—
467C2—241no—
468C2#560111yescharacteristic
469C2#561311yescharacteristic
470C2#561711yescharacteristic
471C2#562011yescharacteristic
472C2#580511yescharacteristic
473C2#580611yescharacteristic
474C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2714
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#898C2semidirect
3. #5223C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #5221C2semidirect
8. #5221C2semidirect
9. #5213C2semidirect
10. #5210C2semidirect
11. #4980C2semidirect
12. #4961C2semidirect
13. #4957C2semidirect
14. #4953C2semidirect
15. #4944C2semidirect
55. C2×#121D3semidirect
105. C3×C4⋊C4D4semidirect
273. C4⋊C4#67semidirect
463. C3#2714semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#8982
#52232
#52222
#52212
#52132
#52102
#49802
#49612
#49572
#49532
#49442
#27143