Finite Group Databasebeta

← #72222621

Subgroups — #72222621

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
1C3×#70885C211yesmaximal, characteristic
2#11777359C211yesmaximal, characteristic, complemented
3#11777340C211yesmaximal, characteristic, complemented
4#11772183C211yesmaximal, characteristic, complemented
5#11771610C211yesmaximal, characteristic, complemented
6C3×#67762C211yesmaximal, characteristic, complemented
7#11771331C211yesmaximal, characteristic, complemented
8#10707072—31nomaximal
9#10707069—31nomaximal
10C32×C4×D8C2211yescharacteristic
11D4×C3⋊(C3⋊C8)—21no—
12C2×C3×#826C411yescharacteristic
13C3×C4×D24C2211yescharacteristic
14#10590329—21no—
15D4⋊(C3⋊(C3⋊C8))—21no—
16#10589323C2211yescharacteristic
17#10589321C2211yescharacteristic
18C3×#4562C411yescharacteristic
19C3×#4491C411yescharacteristic
20C3×#4490C411yescharacteristic
21C3×#4461C2211yescharacteristic
22C3×#4461C2211yescharacteristic
23C3×#4457C2211yescharacteristic
24#10589080—21no—
25C3×#2177—31no—
26#70885D311yescharacteristic
27#70878—31no—
28#70878—31no—
29#70878—31no—
30#70878—61no—
31#68518—31no—
32C3×#1333—31no—
33#67762—31no—
34#67762D311yescharacteristic, complemented
35#67516—31no—
36#67513—31no—
37#67512—31no—
38#67511—31no—
39#67464—31no—
40C2×C32×D8C2×C411yescharacteristic
41C32×C4×D4—21no—
42C22×C3⋊(C3⋊C8)—41no—
43C2×C3×D24C2×C411yescharacteristic
44C3×C4×D12—21no—
45C2×D3×C3⋊C8—41no—
46C32×#107C2×C411yescharacteristic
47C32×#102—21no—
48C32×C4×C8C2311yescharacteristic
49#63657—41no—
50C4⋊(C3⋊(C3⋊C8))—21no—
51C4×C3⋊(C3⋊C8)D411yescharacteristic
52C4×C3⋊(C3⋊C8)D411yescharacteristic
53C3×#826C814yescomplemented
54C3×#821—21no—
55C3×#818C2×C411yescharacteristic
56C2×C3×C3⋊C16—21no—
57C2×C3×C3⋊C16C2×C411yescharacteristic
58C2×C3×C3⋊C16C2×C411yescharacteristic
59C3×C4⋊(C3⋊C8)—21no—
60(C2×D3)⋊(C3⋊C8)—41no—
61#12067—91noSylow
62C2×C3×D16—31no—
63C3×C4×D8—21no—
64C3×C4×D8—31no—
65C3×C4×D8C2×D311yescharacteristic
66C3×C4×D8C2×D311yescharacteristic
67C2×#826Dic311yescharacteristic
68D4×C3⋊C8—61no—
69D4×C3⋊C8—61no—
70D4×C3⋊C8—61no—
71D4×C3⋊C8—62no—
72#4834—61no—
73C4×D24C2×D311yescharacteristic
74C3×#367—31no—
75C3×#357—31no—
76C3×#357—31no—
77C3×#346—31no—
78C3×#346—31no—
79C3×#334—31no—
80#4562Dic311yescharacteristic
81D4⋊(C3⋊C8)—61no—
82D4⋊(C3⋊C8)—61no—
83D4⋊(C3⋊C8)—61no—
84D4⋊(C3⋊C8)—62no—
85#4491Dic311yescharacteristic
86#4490Dic311yescharacteristic
87#4483—61no—
88#4461C2×D311yescharacteristic
89#4461C2×D311yescharacteristic
90#4457—31no—
91#4457—31no—
92#4457—61no—
93#4457C2×D311yescharacteristic
94#4455—31no—
95#4455—31no—
96#4455—61no—
97C2×C32×D4—21no—
98C22×C32×C4—41no—
99C2×C3×D12—21no—
100C2×C3×C4×D3—41no—
101C32×D8—21no—
102C32×D8C2×C812yescomplemented
103C2×C32×C8C22×C411yescharacteristic
104C2×C32×C8#4111yescharacteristic
105C32×C4⋊C4—21no—
106C32×C22⋊C4—41no—
107C32×C42C2×D411yescharacteristic
108C2×C3⋊(C3⋊C8)—21no—
109C2×C3⋊(C3⋊C8)—21no—
110C2×C3⋊(C3⋊C8)—41no—
111C2×C3⋊(C3⋊C8)—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—31no—
122#1580—91no—
123#1580—91no—
124#1368—91no—
125#1333—31no—
126#1333—91no—
127#1313—91no—
128C2×C3×D8—21no—
129C2×C3×D8—31no—
130C2×C3×D8C4×D311yescharacteristic
131C2×C3×D8C2×Dic311yescharacteristic
132C3×C4×D4—22no—
133C3×C4×D4—21no—
134C3×C4×D4—21no—
135C3×C4×D4—61no—
136C22×C3⋊C8—121no—
137C22×C3⋊C8—122no—
138C22×C3⋊C8—121no—
139C22×C3⋊C8—121no—
140C2×D24C2×Dic311yescharacteristic
141C2×D3×C8—121no—
142C4×D12—21no—
143C3×D16—34no—
144C2×C3×C16—31no—
145C2×C3×C16—31no—
146C2×C3×C16—61no—
147C3×#107—21no—
148C3×#107—31no—
149C3×#107C4×D311yescharacteristic
150C3×#107C2×Dic311yescharacteristic
151C3×C4⋊C8—61no—
152C3×#102—22no—
153C3×#102—21no—
154C3×#102—21no—
155C3×#102—61no—
156C3×C4×C8—21no—
157C3×C4×C8C22×D311yescharacteristic
158C3×C4×C8C22×D311yescharacteristic
159#830—121no—
160#830—122no—
161#830—121no—
162#830—121no—
163#826C3⋊C814yescomplemented
164#821—21no—
165(C2×D3)⋊C8—121no—
166#818C2×Dic311yescharacteristic
167C2×C3⋊C16—21no—
168C2×C3⋊C16C2×Dic311yescharacteristic
169C2×C3⋊C16C2×Dic311yescharacteristic
170C4⋊(C3⋊C8)—21no—
171C4⋊(C3⋊C8)—62no—
172C4⋊(C3⋊C8)—61no—
173C4⋊(C3⋊C8)—61no—
174C4×C3⋊C8—31no—
175C4×C3⋊C8—31no—
176C4×C3⋊C8—31no—
177C4×C3⋊C8—31no—
178C4×C3⋊C8—61no—
179C4×C3⋊C8—61no—
180C23×C32—41no—
181C22×C3×D3—41no—
182C32×D4—22no—
183C32×D4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—81no—
186C2×C32×C4—81no—
187C2×C32×C4C4×D411yescharacteristic
188C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
190C2×C3×Dic3—41no—
191C3×D12—22no—
192C3×D12—41no—
193C3×C4×D3—81no—
194C3×C4×D3—81no—
195C32×C8—21no—
196C32×C8C22×C811yescharacteristic
197C32×C8#13111yescharacteristic
198C3⋊(C3⋊C8)—41no—
199C3⋊(C3⋊C8)—41no—
200C3⋊(C3⋊C8)—81no—
201C3×C3⋊C8—81no—
202C2×D16—31no—
203C4×D8—31no—
204C4×D8D3211yescharacteristic
205C8×D4—181no—
206C8×D4—181no—
207#367—31no—
208#357—31no—
209#357—31no—
210#346—31no—
211#346—31no—
212#334—31no—
213#334—91no—
214D4⋊C8—181no—
215D4⋊C8—181no—
216#322—91no—
217C2×C3×D4—22no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—61no—
221C22×C3×C4—42no—
222C22×C3×C4—41no—
223C22×C3×C4—41no—
224C22×C3×C4—121no—
225C2×D12—21no—
226C2×C4×D3—41no—
227C3×D8—22no—
228C3×D8—21no—
229C3×D8—21no—
230C3×D8—22no—
231C3×D8—32no—
232C3×D8—61no—
233C3×D8D3×C812yescomplemented
234C3×D8C2×C3⋊C812yescomplemented
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—121no—
238C2×C3×C8C2×C4×D311yescharacteristic
239C2×C3×C8#23711yescharacteristic
240C2×C3×C8#23711yescharacteristic
241C2×C3×C8C22×Dic311yescharacteristic
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—61no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—121no—
250C3×C42—21no—
251C3×C42D3×D411yescharacteristic
252C3×C42D3×D411yescharacteristic
253(C2×D3)⋊C4—41no—
254C4⋊Dic3—21no—
255C2×C3⋊C8—41no—
256C2×C3⋊C8—62no—
257C2×C3⋊C8—61no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—121no—
262C2×C3⋊C8—122no—
263C2×C3⋊C8—121no—
264C2×C3⋊C8—121no—
265C2×C3⋊C8—242no—
266C2×C3⋊C8—244no—
267C2×C3⋊C8—242no—
268C2×C3⋊C8—242no—
269D24—21no—
270D24C2×C3⋊C812yescomplemented
271D3×C8—242no—
272C3×C16—32no—
273C3×C16—61no—
274C3×C16—121no—
275C3⋊C16—21no—
276C3⋊C16—41no—
277C3⋊C16C2×C3⋊C812yescomplemented
278C22×C32—42no—
279C22×C32—81no—
280C22×C32—81no—
281C22×C32#44211yescharacteristic
282C2×C3×D3—42no—
283C2×C3×D3—81no—
284C2×C3×D3—81no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—81no—
288C32×C4C8×D411yescharacteristic
289C32×C4#44411yescharacteristic
290C3×Dic3—81no—
291C2×D8—31no—
292C2×D8D3×Dic311yescharacteristic
293C22×C8—361no—
294C22×C8—361no—
295C4×D4—21no—
296C4×D4—61no—
297D16—34no—
298C2×C16—31no—
299C2×C16—31no—
300C2×C16—61no—
301#107—31no—
302#107D3×Dic311yescharacteristic
303C4⋊C8—61no—
304C4⋊C8—181no—
305#102—21no—
306#102—61no—
307C22⋊C8—361no—
308C22⋊C8—361no—
309C4×C8—91no—
310C4×C8—91no—
311C4×C8C2×D3211yescharacteristic
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—121no—
316C22×D3—41no—
317C3×D4—24no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—62no—
324C3×D4—121no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—82no—
332C2×C3×C4—81no—
333C2×C3×C4—81no—
334C2×C3×C4—82no—
335C2×C3×C4—81no—
336C2×C3×C4—81no—
337C2×C3×C4—121no—
338C2×C3×C4—241no—
339C2×C3×C4—241no—
340C2×C3×C4#67⋊C411yescharacteristic
341C2×C3×C4#91111yescharacteristic
342C2×C3×C4#91111yescharacteristic
343C2×C3×C4#91111yescharacteristic
344C2×C3×C4#91411yescharacteristic
345C2×C3×C4D4×Dic311yescharacteristic
346C2×Dic3—41no—
347D12—22no—
348D12—41no—
349C4×D3—81no—
350C4×D3—81no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—21no—
355C3×C8—41no—
356C3×C8—241no—
357C3×C8C2×D3×C811yescharacteristic
358C3×C8#90711yescharacteristic
359C3×C8C22×C3⋊C811yescharacteristic
360C3×C8#94811yescharacteristic
361C3⋊C8—81no—
362C3⋊C8—121no—
363C3⋊C8—122no—
364C3⋊C8—121no—
365C3⋊C8—121no—
366C3⋊C8—121no—
367C3⋊C8—241no—
368C3⋊C8—242no—
369C3⋊C8—241no—
370C3⋊C8—241no—
371C2×C32—82no—
372C2×C32#158011yescharacteristic
373C2×C32AGL(1,ℤ/16)11yescharacteristic
374C2×C32AGL(1,ℤ/16)11yescharacteristic
375C3×D3—82no—
376C2×D4—21no—
377C2×D4—61no—
378C22×C4—41no—
379C22×C4—121no—
380D8—21no—
381D8—32no—
382D8—61no—
383D8D3×C3⋊C812yescomplemented
384C2×C8—121no—
385C2×C8—181no—
386C2×C8—181no—
387C2×C8—361no—
388C2×C8—722no—
389C2×C8—722no—
390C2×C8C2×D3×Dic311yescharacteristic
391C2×C8#380511yescharacteristic
392C4⋊C4—21no—
393C4⋊C4—61no—
394C22⋊C4—41no—
395C22⋊C4—121no—
396C42#381111yescharacteristic
397C16—32no—
398C16—61no—
399C16—121no—
400C22×C3—21no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—122no—
411C22×C3—241no—
412C22×C3—241no—
413C22×C3#478511yescharacteristic
414C22×C3#515211yescharacteristic
415C2×D3—42no—
416C2×D3—81no—
417C2×D3—81no—
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—82no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—241no—
430C3×C4#67⋊C811yescharacteristic
431C3×C4#482311yescharacteristic
432C3×C4D4×C3⋊C811yescharacteristic
433C3×C4#519711yescharacteristic
434Dic3—81no—
435C32#1206711yescharacteristic, complemented, Sylow
436C23—41no—
437C23—121no—
438D4—22no—
439D4—41no—
440D4—62no—
441D4—121no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—121no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4#6392711yescharacteristic
449C2×C4#6393711yescharacteristic
450C2×C4#67⋊Dic311yescharacteristic
451C8—21no—
452C8—241no—
453C8—361no—
454C8—361no—
455C8—721no—
456C8C2×D3×C3⋊C811yescharacteristic
457C8#6381511yescharacteristic
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—84no—
462C2×C3—82no—
463C2×C3—82no—
464C2×C3—242no—
465C2×C3#6815111yescharacteristic
466C2×C3#6924911yescharacteristic
467C2×C3#6925411yescharacteristic
468C2×C3#7087811yescharacteristic
469C2×C3#7151511yescharacteristic
470C2×C3#7151511yescharacteristic
471D3—82no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—122no—
476C22—241no—
477C22—241no—
478C22#1059143611yescharacteristic
479C4—21no—
480C4—21no—
481C4—81no—
482C4—241no—
483C4#67⋊(C3⋊C8)11yescharacteristic
484C4#1059167611yescharacteristic
485C3—21no—
486C3#1070706911yescharacteristic, complemented
487C3#1070707211yescharacteristic, complemented
488C2—82no—
489C2—242no—
490C2#1177735211yescharacteristic
491C2#1179797011yescharacteristic
492C2#1179797911yescharacteristic
subgroupG/H
centerC22#10591436
commutatorC32×C8C22×C8
FrattiniC4×C8C2×D32
FittingC32×C4×D8C22
socleC22×C32#442
prime radicalC2×C3×#826C4
cosocleC32×C4×C8C23
hypercenterC4×D8D32
hypocenterC32#12067
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. #11777359C2semidirect
3. #11777340C2semidirect
4. #11772183C2semidirect
5. #11771610C2semidirect
6. C3×#67762C2semidirect
7. #11771331C2semidirect
34. #67762D3semidirect
53. C3×#826C8semidirect
102. C32×D8C2×C8semidirect
116. C3×D24C2×C8semidirect
120. C3×C3⋊C16C2×C8semidirect
163. #826C3⋊C8semidirect
233. C3×D8D3×C8semidirect
234. C3×D8C2×C3⋊C8semidirect
270. D24C2×C3⋊C8semidirect
277. C3⋊C16C2×C3⋊C8semidirect
383. D8D3×C3⋊C8semidirect
435. C32#12067semidirect
486. C3#10707069semidirect
487. C3#10707072semidirect

Maximal subgroups

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

subgroupindex
C3×#708852
#117773592
#117773402
#117721832
#117716102
C3×#677622
#117713312
#107070723
#107070693