Finite Group Databasebeta

← #72222618

Subgroups — #72222618

Order 2304.

588 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11797822C211yesmaximal, characteristic
2C3×#68077C211yesmaximal, characteristic, complemented
3#11777343C211yesmaximal, characteristic, complemented
4#11772183C211yesmaximal, characteristic, complemented
5#11771611C211yesmaximal, characteristic, complemented
6#11771610C211yesmaximal, characteristic, complemented
7C3×#67442C211yesmaximal, characteristic, complemented
8#10707073—31nomaximal
9#10707070—31nomaximal
10C4×#3745C2211yescharacteristic
11C8×C4⋊(C3⋊D3)—21no—
12C3×C4×D24C2211yescharacteristic
13C3×C8×D12—21no—
14C2×#63546C411yescharacteristic
15C32×#322C2211yescharacteristic
16#10589325—21no—
17#10589323C2211yescharacteristic
18C3×#4461C2211yescharacteristic
19C3×#4461C2211yescharacteristic
20C3×#4459—21no—
21C3×#4457C2211yescharacteristic
22#10589141C411yescharacteristic
23#10589116C411yescharacteristic
24#10589115C411yescharacteristic
25C3×#1580—31no—
26#70885—31no—
27#69078—31no—
28#68077—31no—
29#68077—31no—
30#68077—61no—
31#68077D311yescharacteristic, complemented
32C3×#1313—31no—
33#67762—31no—
34#67516—31no—
35#67516—31no—
36#67513—31no—
37#67465—31no—
38#67464—31no—
39#67442D311yescharacteristic, complemented
40C2×#3745C2×C411yescharacteristic
41C2×C8×C3⋊D3—41no—
42C4×C4⋊(C3⋊D3)—21no—
43C2×C3×D24C2×C411yescharacteristic
44C2×C3×D3×C8—41no—
45C3×C4×D12—21no—
46C32×C4×C8D411yescharacteristic
47C32×C4×C8D411yescharacteristic
48C32×C4×C8C2311yescharacteristic
49#63648—21no—
50#63647—41no—
51#63645C2×C411yescharacteristic
52C4⋊(C3⋊(C3⋊C8))—21no—
53C3×#821—21no—
54C3×(C2×D3)⋊C8—41no—
55C3×#818C2×C411yescharacteristic
56C2×C3×C3⋊C16—21no—
57C2×C3×C3⋊C16C2×C411yescharacteristic
58C2×C3×C3⋊C16C2×C411yescharacteristic
59C3×C4⋊(C3⋊C8)—21no—
60#63546C814yescomplemented
61#12067—91noSylow
62C3×C4×D8—31no—
63C3×C8×D4—61no—
64C2×#826—31no—
65C2×D48—31no—
66C4×D24—31no—
67C4×D24—31no—
68C4×D24—61no—
69C4×D24C2×D311yescharacteristic
70C8×D12—21no—
71C8×D12—62no—
72C8×D12—61no—
73C8×D12—61no—
74C3×#346—31no—
75C3×#346—31no—
76C3×#334—31no—
77C3×D4⋊C8—61no—
78C3×#322—21no—
79C3×#322C2×D311yescharacteristic
80C3×#322C2×D311yescharacteristic
81#4562—31no—
82#4509—31no—
83#4509—31no—
84#4505—31no—
85#4491—31no—
86#4490—31no—
87#4461C2×D311yescharacteristic
88#4461C2×D311yescharacteristic
89#4459—21no—
90#4459—62no—
91#4459—61no—
92#4459—61no—
93#4457—31no—
94#4457—31no—
95#4457—61no—
96#4457C2×D311yescharacteristic
97C2×C4⋊(C3⋊D3)—21no—
98C2×C4×C3⋊D3—41no—
99C2×C3×D12—21no—
100C2×C3×C4×D3—41no—
101C2×C32×C8—21no—
102C2×C32×C8—21no—
103C2×C32×C8C22×C411yescharacteristic
104C2×C32×C8#4111yescharacteristic
105C32×C42C2×D411yescharacteristic
106#3752—41no—
107C4⋊(C3⋊Dic3)—21no—
108C2×C3⋊(C3⋊C8)—41no—
109#3745—21no—
110#3745C2×C812yescomplemented
111C8×C3⋊D3—82no—
112C3×(C2×D3)⋊C4—41no—
113C3×C4⋊Dic3—21no—
114C2×C3×C3⋊C8—41no—
115C3×D24—21no—
116C3×D24C2×C812yescomplemented
117C3×D3×C8—82no—
118C3×C3⋊C16—21no—
119C3×C3⋊C16—41no—
120C3×C3⋊C16C2×C812yescomplemented
121#2177—91no—
122#1580—31no—
123#1580—91no—
124#1368—91no—
125#1333—91no—
126#1333—91no—
127#1313—31no—
128C2×C3×D8—31no—
129C22×C3×C8—121no—
130C3×C4×D4—61no—
131C2×D24—31no—
132C2×D24—31no—
133C2×D24—61no—
134C2×D24C4×D311yescharacteristic
135C2×D3×C8—41no—
136C2×D3×C8—122no—
137C2×D3×C8—121no—
138C2×D3×C8—121no—
139C4×D12—21no—
140C4×D12—62no—
141C4×D12—61no—
142C4×D12—61no—
143C2×C3×C16—31no—
144C2×C3×C16—31no—
145C2×C3×C16—61no—
146C3×#107—31no—
147C3×C4⋊C8—61no—
148C3×#102—61no—
149C3×C22⋊C8—121no—
150C3×C4×C8—21no—
151C3×C4×C8—21no—
152C3×C4×C8—21no—
153C3×C4×C8D1211yescharacteristic
154C3×C4×C8D1211yescharacteristic
155C3×C4×C8#6711yescharacteristic
156C3×C4×C8#6711yescharacteristic
157C3×C4×C8C22×D311yescharacteristic
158C3×C4×C8C22×D311yescharacteristic
159#826—34no—
160#821—21no—
161#821—62no—
162#821—61no—
163#821—61no—
164(C2×D3)⋊C8—41no—
165(C2×D3)⋊C8—122no—
166(C2×D3)⋊C8—121no—
167(C2×D3)⋊C8—121no—
168#818—31no—
169#818—31no—
170#818—61no—
171#818C4×D311yescharacteristic
172C2×C3⋊C16—21no—
173C2×C3⋊C16C4×D311yescharacteristic
174C2×C3⋊C16C4×D311yescharacteristic
175C4⋊(C3⋊C8)—21no—
176C4⋊(C3⋊C8)—62no—
177C4⋊(C3⋊C8)—61no—
178C4⋊(C3⋊C8)—61no—
179D48—34no—
180C22×C3⋊D3—41no—
181C22×C3×D3—41no—
182C2×C32×C4C4×D411yescharacteristic
183C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
184C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
185C2×C3⋊Dic3—41no—
186C4⋊(C3⋊D3)—22no—
187C4⋊(C3⋊D3)—41no—
188C4×C3⋊D3—81no—
189C4×C3⋊D3—81no—
190C2×C3×Dic3—41no—
191C3×D12—22no—
192C3×D12—41no—
193C3×C4×D3—81no—
194C3×C4×D3—81no—
195C32×C8—21no—
196C32×C8—41no—
197C32×C8—41no—
198C32×C8C22×C811yescharacteristic
199C32×C8#13111yescharacteristic
200C3⋊(C3⋊C8)—81no—
201C3×C3⋊C8—81no—
202C2×D16—91no—
203C4×D8—31no—
204C4×D8—91no—
205C8×D4—61no—
206C8×D4—181no—
207#367—91no—
208#357—91no—
209#357—91no—
210#346—31no—
211#346—31no—
212#334—31no—
213#334—91no—
214D4⋊C8—61no—
215D4⋊C8—181no—
216#322D3211yescharacteristic
217C2×C3×D4—61no—
218C22×C3×C4—121no—
219C2×D12—21no—
220C2×D12—62no—
221C2×D12—61no—
222C2×D12—61no—
223C2×C4×D3—41no—
224C2×C4×D3—122no—
225C2×C4×D3—121no—
226C2×C4×D3—121no—
227C3×D8—32no—
228C3×D8—61no—
229C2×C3×C8—22no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—41no—
237C2×C3×C8—121no—
238C2×C3×C8—242no—
239C2×C3×C8C2×C4×D311yescharacteristic
240C2×C3×C8C2×C4×D311yescharacteristic
241C2×C3×C8#23511yescharacteristic
242C2×C3×C8#23511yescharacteristic
243C3×C4⋊C4—61no—
244C3×C22⋊C4—121no—
245C3×C42—21no—
246C3×C42C2×D1211yescharacteristic
247C3×C42C2×#6711yescharacteristic
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—122no—
250(C2×D3)⋊C4—121no—
251(C2×D3)⋊C4—121no—
252C4⋊Dic3—21no—
253C4⋊Dic3—62no—
254C4⋊Dic3—61no—
255C4⋊Dic3—61no—
256C2×C3⋊C8—41no—
257C2×C3⋊C8—122no—
258C2×C3⋊C8—121no—
259C2×C3⋊C8—121no—
260D24—21no—
261D24—32no—
262D24—32no—
263D24—62no—
264D24—62no—
265D24—61no—
266D24—61no—
267D24D3×C812yescomplemented
268D3×C8—82no—
269D3×C8—244no—
270D3×C8—242no—
271D3×C8—242no—
272C3×C16—32no—
273C3×C16—61no—
274C3×C16—121no—
275C3⋊C16—21no—
276C3⋊C16—41no—
277C3⋊C16D3×C812yescomplemented
278C22×C32#44211yescharacteristic
279C2×C3⋊D3—42no—
280C2×C3⋊D3—81no—
281C2×C3⋊D3—81no—
282C2×C3×D3—42no—
283C2×C3×D3—81no—
284C2×C3×D3—81no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4C8×D411yescharacteristic
288C32×C4#44411yescharacteristic
289C3⋊Dic3—81no—
290C3×Dic3—81no—
291C2×D8—31no—
292C2×D8—91no—
293C22×C8—121no—
294C22×C8—361no—
295C4×D4—61no—
296C4×D4—181no—
297D16—94no—
298C2×C16—31no—
299C2×C16—31no—
300C2×C16—61no—
301#107—31no—
302#107—91no—
303C4⋊C8—61no—
304C4⋊C8—181no—
305#102—61no—
306#102—181no—
307C22⋊C8—121no—
308C22⋊C8—361no—
309C4×C8#62611yescharacteristic
310C4×C8#62611yescharacteristic
311C4×C8C2×D3211yescharacteristic
312C23×C3—121no—
313C22×D3—41no—
314C22×D3—122no—
315C22×D3—121no—
316C22×D3—121no—
317C3×D4—62no—
318C3×D4—121no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—121no—
323C2×C3×C4—241no—
324C2×C3×C4—241no—
325C2×C3×C4C4×D1211yescharacteristic
326C2×C3×C4#90811yescharacteristic
327C2×C3×C4#90811yescharacteristic
328C2×C3×C4C4×#6711yescharacteristic
329C2×C3×C4#93211yescharacteristic
330C2×C3×C4#94911yescharacteristic
331C2×Dic3—41no—
332C2×Dic3—122no—
333C2×Dic3—121no—
334C2×Dic3—121no—
335D12—22no—
336D12—41no—
337D12—64no—
338D12—62no—
339D12—62no—
340D12—122no—
341D12—121no—
342D12—121no—
343C4×D3—81no—
344C4×D3—81no—
345C4×D3—242no—
346C4×D3—241no—
347C4×D3—241no—
348C4×D3—242no—
349C4×D3—241no—
350C4×D3—241no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—21no—
355C3×C8—41no—
356C3×C8—41no—
357C3×C8—42no—
358C3×C8—41no—
359C3×C8—41no—
360C3×C8—41no—
361C3×C8—81no—
362C3×C8—241no—
363C3×C8C2×D3×C811yescharacteristic
364C3×C8C2×D3×C811yescharacteristic
365C3×C8#90111yescharacteristic
366C3×C8#90111yescharacteristic
367C3⋊C8—81no—
368C3⋊C8—242no—
369C3⋊C8—241no—
370C3⋊C8—241no—
371C2×C32#158011yescharacteristic
372C2×C32AGL(1,ℤ/16)11yescharacteristic
373C2×C32AGL(1,ℤ/16)11yescharacteristic
374C3⋊D3—82no—
375C3×D3—82no—
376C2×D4—61no—
377C2×D4—181no—
378C22×C4—121no—
379C22×C4—361no—
380D8—32no—
381D8—61no—
382D8—92no—
383D8—181no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—121no—
387C2×C8—242no—
388C2×C8—361no—
389C2×C8—722no—
390C2×C8#379811yescharacteristic
391C2×C8C4×D3211yescharacteristic
392C4⋊C4—61no—
393C4⋊C4—181no—
394C22⋊C4—121no—
395C22⋊C4—361no—
396C42C2×#62611yescharacteristic
397C16—32no—
398C16—61no—
399C16—121no—
400C22×C3—21no—
401C22×C3—122no—
402C22×C3—241no—
403C22×C3—241no—
404C22×C3#471111yescharacteristic
405C22×C3#501411yescharacteristic
406C2×D3—42no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—124no—
410C2×D3—122no—
411C2×D3—122no—
412C2×D3—242no—
413C2×D3—242no—
414C2×D3—241no—
415C2×D3—241no—
416C2×D3—241no—
417C2×D3—241no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—241no—
427C3×C4C8×D1211yescharacteristic
428C3×C4#471711yescharacteristic
429C3×C4C8×#6711yescharacteristic
430C3×C4#514511yescharacteristic
431Dic3—81no—
432Dic3—242no—
433Dic3—241no—
434Dic3—241no—
435C32#1206711yescharacteristic, complemented, Sylow
436C23—121no—
437C23—361no—
438D4—62no—
439D4—121no—
440D4—182no—
441D4—361no—
442C2×C4—121no—
443C2×C4—241no—
444C2×C4—241no—
445C2×C4—361no—
446C2×C4—721no—
447C2×C4—721no—
448C2×C4#6382511yescharacteristic
449C2×C4#6383011yescharacteristic
450C2×C4C4×#62611yescharacteristic
451C8—21no—
452C8—41no—
453C8—41no—
454C8—241no—
455C8—721no—
456C8D32×C811yescharacteristic
457C8#6380311yescharacteristic
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—242no—
462C2×C3#6807711yescharacteristic
463C2×C3#6909111yescharacteristic
464C2×C3#6909111yescharacteristic
465C2×C3#7084011yescharacteristic
466C2×C3#7091811yescharacteristic
467C2×C3#7092911yescharacteristic
468D3—82no—
469D3—244no—
470D3—242no—
471D3—242no—
472C22—122no—
473C22—241no—
474C22—241no—
475C22—362no—
476C22—721no—
477C22—721no—
478C22#1059034611yescharacteristic
479C4—21no—
480C4—21no—
481C4—241no—
482C4—721no—
483C4C8×#62611yescharacteristic
484C4#1059092411yescharacteristic
485C3—21no—
486C3#1070707011yescharacteristic, complemented
487C3#1070707311yescharacteristic, complemented
488C2—242no—
489C2—722no—
490C2#1177735411yescharacteristic
491C2#1179797211yescharacteristic
492C2#1179797311yescharacteristic
subgroupG/H
centerC22#10590346
commutatorC32×C8C22×C8
FrattiniC4×C8C2×D32
FittingC32×#322C22
socleC22×C32#442
prime radicalC2×#63546C4
cosocleC32×C4×C8C23
hypercenter#322D32
hypocenterC32#12067
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #322

Complements

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

NH ≅ G/Ntype
2. C3×#68077C2semidirect
3. #11777343C2semidirect
4. #11772183C2semidirect
5. #11771611C2semidirect
6. #11771610C2semidirect
7. C3×#67442C2semidirect
31. #68077D3semidirect
39. #67442D3semidirect
60. #63546C8semidirect
110. #3745C2×C8semidirect
116. C3×D24C2×C8semidirect
120. C3×C3⋊C16C2×C8semidirect
267. D24D3×C8semidirect
277. C3⋊C16D3×C8semidirect
435. C32#12067semidirect
486. C3#10707070semidirect
487. C3#10707073semidirect

Maximal subgroups

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

subgroupindex
#117978222
C3×#680772
#117773432
#117721832
#117716112
#117716102
C3×#674422
#107070733
#107070703