Finite Group Databasebeta

← #72222611

Subgroups — #72222611

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
2#11777359C211yesmaximal, characteristic, complemented
3#11777337C211yesmaximal, characteristic, complemented
4C3×#67464C211yesmaximal, characteristic, complemented
5#11771611C211yesmaximal, characteristic, complemented
6C3×#67465C211yesmaximal, characteristic, complemented
7#11771331C211yesmaximal, characteristic, complemented
8#10707071—31nomaximal
9#10707069—31nomaximal
10#10592533—21no—
11C4×#3745C2211yescharacteristic
12C3×C4×D24C2211yescharacteristic
13C2×#63546C411yescharacteristic
14#10590329—21no—
15C32×#334C2211yescharacteristic
16#10589349—21no—
17#10589321C2211yescharacteristic
18C3×#4461C2211yescharacteristic
19C3×#4461C2211yescharacteristic
20C3×#4457C2211yescharacteristic
21#10589141C411yescharacteristic
22#10589116C411yescharacteristic
23#10589115C411yescharacteristic
24#10589080—21no—
25#70885—31no—
26#70878—31no—
27#69078—31no—
28#68518—31no—
29#68518—31no—
30#68518—31no—
31#68518—61no—
32C3×#1368—31no—
33C3×#1333—31no—
34#67516—31no—
35#67512—31no—
36#67511—31no—
37#67465—31no—
38#67465D311yescharacteristic, complemented
39#67464D311yescharacteristic, complemented
40C2×#3745C2×C411yescharacteristic
41C2×C8×C3⋊D3—41no—
42C4×C4⋊(C3⋊D3)—21no—
43C2×C3×D24C2×C411yescharacteristic
44C3×C4×D12—21no—
45C2×D3×C3⋊C8—41no—
46C32×C4⋊C8—21no—
47C32×C4×C8C2311yescharacteristic
48#63648—21no—
49#63647—41no—
50#63645C2×C411yescharacteristic
51C4×C3⋊(C3⋊C8)D411yescharacteristic
52C4×C3⋊(C3⋊C8)D411yescharacteristic
53C3×#821—21no—
54C3×#818C2×C411yescharacteristic
55C2×C3×C3⋊C16—21no—
56C2×C3×C3⋊C16C2×C411yescharacteristic
57C2×C3×C3⋊C16C2×C411yescharacteristic
58C3×C4⋊(C3⋊C8)—21no—
59(C2×D3)⋊(C3⋊C8)—41no—
60#63546C814yescomplemented
61#12067—91noSylow
62C3×C4×D8—31no—
63C2×#826—31no—
64D4×C3⋊C8—61no—
65C2×D48—31no—
66#4834—61no—
67#4834—61no—
68#4834—61no—
69#4834—62no—
70C4×D24—31no—
71C4×D24—31no—
72C4×D24—61no—
73C4×D24C2×D311yescharacteristic
74C3×#346—31no—
75C3×#346—31no—
76C3×#334—21no—
77C3×#334—31no—
78C3×#334C2×D311yescharacteristic
79C3×#334C2×D311yescharacteristic
80#4562—31no—
81D4⋊(C3⋊C8)—61no—
82#4509—31no—
83#4509—31no—
84#4505—31no—
85#4491—31no—
86#4490—31no—
87#4483—61no—
88#4483—61no—
89#4483—61no—
90#4483—62no—
91#4461C2×D311yescharacteristic
92#4461C2×D311yescharacteristic
93#4457C2×D311yescharacteristic
94#4455—31no—
95#4455—31no—
96#4455—61no—
97C2×C4⋊(C3⋊D3)—21no—
98C2×C4×C3⋊D3—41no—
99C2×C3×D12—21no—
100C2×C3×C4×D3—41no—
101C2×C32×C8—41no—
102C2×C32×C8C22×C411yescharacteristic
103C2×C32×C8#4111yescharacteristic
104C32×C42C2×D411yescharacteristic
105#3752—41no—
106C4⋊(C3⋊Dic3)—21no—
107C2×C3⋊(C3⋊C8)—21no—
108C2×C3⋊(C3⋊C8)—21no—
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
117D3×C3⋊C8—82no—
118C3×C3⋊C16—21no—
119C3×C3⋊C16—41no—
120C3×C3⋊C16C2×C812yescomplemented
121#2177—91no—
122#1580—91no—
123#1580—91no—
124#1368—31no—
125#1333—31no—
126#1333—91no—
127#1313—91no—
128C2×C3×D8—31no—
129C3×C4×D4—61no—
130C22×C3⋊C8—121no—
131C2×D24—31no—
132C2×D24—31no—
133C2×D24—61no—
134C2×D24C4×D311yescharacteristic
135C2×D3×C8—122no—
136C2×D3×C8—121no—
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—22no—
148C3×C4⋊C8—21no—
149C3×C4⋊C8—21no—
150C3×C4⋊C8—61no—
151C3×#102—61no—
152C3×C4×C8—21no—
153C3×C4×C8C22×D311yescharacteristic
154C3×C4×C8C22×D311yescharacteristic
155#830—121no—
156#826—34no—
157#821—21no—
158#821—62no—
159#821—61no—
160#821—61no—
161(C2×D3)⋊C8—122no—
162(C2×D3)⋊C8—121no—
163(C2×D3)⋊C8—121no—
164(C2×D3)⋊C8—121no—
165#818—31no—
166#818—31no—
167#818—61no—
168#818C4×D311yescharacteristic
169C2×C3⋊C16—21no—
170C2×C3⋊C16C4×D311yescharacteristic
171C2×C3⋊C16C4×D311yescharacteristic
172C4⋊(C3⋊C8)—21no—
173C4×C3⋊C8—31no—
174C4×C3⋊C8—31no—
175C4×C3⋊C8—31no—
176C4×C3⋊C8—31no—
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—81no—
197C32×C8C22×C811yescharacteristic
198C32×C8#13111yescharacteristic
199C3⋊(C3⋊C8)—41no—
200C3⋊(C3⋊C8)—41no—
201C3×C3⋊C8—81no—
202C2×D16—91no—
203C4×D8—31no—
204C4×D8—91no—
205C8×D4—181no—
206C8×D4—181no—
207#367—91no—
208#357—91no—
209#357—91no—
210#346—31no—
211#346—31no—
212#334—31no—
213#334D3211yescharacteristic
214D4⋊C8—181no—
215D4⋊C8—181no—
216#322—91no—
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—21no—
230C2×C3×C8—21no—
231C2×C3×C8—42no—
232C2×C3×C8—41no—
233C2×C3×C8—41no—
234C2×C3×C8—121no—
235C2×C3×C8C2×C4×D311yescharacteristic
236C2×C3×C8C2×C4×D311yescharacteristic
237C2×C3×C8#23711yescharacteristic
238C2×C3×C8#23911yescharacteristic
239C3×C4⋊C4—61no—
240C3×C22⋊C4—121no—
241C3×C42—21no—
242C3×C42D3×D411yescharacteristic
243C3×C42D3×D411yescharacteristic
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—122no—
246(C2×D3)⋊C4—121no—
247(C2×D3)⋊C4—121no—
248C4⋊Dic3—21no—
249C4⋊Dic3—62no—
250C4⋊Dic3—61no—
251C4⋊Dic3—61no—
252C2×C3⋊C8—41no—
253C2×C3⋊C8—62no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8—61no—
257C2×C3⋊C8—61no—
258C2×C3⋊C8—121no—
259C2×C3⋊C8—242no—
260D24—21no—
261D24—32no—
262D24—32no—
263D24—62no—
264D24—62no—
265D24—61no—
266D24—61no—
267D24D3×C812yescomplemented
268D3×C8—244no—
269D3×C8—242no—
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—361no—
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—21no—
304C4⋊C8—61no—
305#102—61no—
306#102—181no—
307C22⋊C8—361no—
308C22⋊C8—361no—
309C4×C8—91no—
310C4×C8—91no—
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×C4#67⋊C411yescharacteristic
326C2×C3×C4#89311yescharacteristic
327C2×C3×C4#91111yescharacteristic
328C2×C3×C4#91411yescharacteristic
329C2×C3×C4#91411yescharacteristic
330C2×C3×C4#91411yescharacteristic
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—82no—
357C3×C8—81no—
358C3×C8—81no—
359C3×C8—241no—
360C3×C8C2×D3×C811yescharacteristic
361C3×C8C2×D3×C811yescharacteristic
362C3×C8#90711yescharacteristic
363C3×C8#90711yescharacteristic
364C3⋊C8—81no—
365C3⋊C8—122no—
366C3⋊C8—121no—
367C3⋊C8—121no—
368C3⋊C8—121no—
369C3⋊C8—121no—
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—41no—
385C2×C8—121no—
386C2×C8—181no—
387C2×C8—181no—
388C2×C8—722no—
389C2×C8—722no—
390C2×C8#379611yescharacteristic
391C2×C8C4×D3211yescharacteristic
392C4⋊C4—61no—
393C4⋊C4—181no—
394C22⋊C4—121no—
395C22⋊C4—361no—
396C42#381111yescharacteristic
397C16—32no—
398C16—61no—
399C16—121no—
400C22×C3—21no—
401C22×C3—122no—
402C22×C3—241no—
403C22×C3—241no—
404C22×C3#478511yescharacteristic
405C22×C3#486911yescharacteristic
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×C4#67⋊C811yescharacteristic
428C3×C4#482311yescharacteristic
429C3×C4#483411yescharacteristic
430C3×C4#489411yescharacteristic
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#6388111yescharacteristic
449C2×C4#6393711yescharacteristic
450C2×C4#6394111yescharacteristic
451C8—21no—
452C8—81no—
453C8—241no—
454C8—361no—
455C8—361no—
456C8D32×C811yescharacteristic
457C8#6380511yescharacteristic
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—242no—
462C2×C3#6815111yescharacteristic
463C2×C3#6851811yescharacteristic
464C2×C3#6924911yescharacteristic
465C2×C3#6925411yescharacteristic
466C2×C3#6931011yescharacteristic
467C2×C3#6931011yescharacteristic
468D3—82no—
469D3—244no—
470D3—242no—
471D3—242no—
472C22—122no—
473C22—241no—
474C22—241no—
475C22—362no—
476C22—721no—
477C22—721no—
478C22#1059046211yescharacteristic
479C4—21no—
480C4—21no—
481C4—241no—
482C4—721no—
483C4#1059090611yescharacteristic
484C4#1059097811yescharacteristic
485C3—21no—
486C3#1070706911yescharacteristic, complemented
487C3#1070707111yescharacteristic, complemented
488C2—242no—
489C2—722no—
490C2#1177734911yescharacteristic
491C2#1179796711yescharacteristic
492C2#1179798211yescharacteristic
subgroupG/H
centerC22#10590462
commutatorC32×C8C22×C8
FrattiniC4×C8C2×D32
FittingC32×#334C22
socleC22×C32#442
prime radicalC2×#63546C4
cosocleC32×C4×C8C23
hypercenter#334D32
hypocenterC32#12067
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #334

Complements

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

NH ≅ G/Ntype
2. #11777359C2semidirect
3. #11777337C2semidirect
4. C3×#67464C2semidirect
5. #11771611C2semidirect
6. C3×#67465C2semidirect
7. #11771331C2semidirect
38. #67465D3semidirect
39. #67464D3semidirect
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#10707069semidirect
487. C3#10707071semidirect

Maximal subgroups

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

subgroupindex
#117978222
#117773592
#117773372
C3×#674642
#117716112
C3×#674652
#117713312
#107070713
#107070693