Finite Group Databasebeta

← #10691273

Subgroups — #10691273

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83349C211yesmaximal, characteristic
2C8×#235C211yesmaximal, characteristic
3C3×#1493C211yesmaximal, characteristic, complemented
4#71835C211yesmaximal, characteristic, complemented
5#70529C211yesmaximal, characteristic
6#67714C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9207—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#825—21no—
14C8×#67—22no—
15#5094C411yescharacteristic, complemented
16#5091C411yescharacteristic, complemented
17#5087C411yescharacteristic, complemented
18#5084C411yescharacteristic, complemented
19#5066—21no—
20#5063—21no—
21#5002—21no—
22#4998—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C8×D12—21no—
27#4685—21no—
28C4×D3×C8—21no—
29C8×Dic6—21no—
30C3×Q8⋊C8—21no—
31C3×Q8⋊C8—21no—
32#4555—21no—
33#4538—22no—
34#3076—31no—
35C8×#41—31no—
36#1806—31no—
37#1503—31no—
38#1493D311yescharacteristic, complemented
39#1301—31no—
40#1279—31no—
41C2×#235D411yescharacteristic
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C42C2311yescharacteristic
53C4×#67—21no—
54C4×#67—21no—
55#926D411yescharacteristic
56C2×C4⋊Dic3—21no—
57C2×C4×Dic3—21no—
58C22×C3⋊C8—21no—
59C2×D3×C8—22no—
60C4×D12C2×C411yescharacteristic
61C4×D12D411yescharacteristic
62#872—21no—
63C42×D3—21no—
64C4×Dic6C2×C411yescharacteristic
65C4×Dic6D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8D411yescharacteristic
70C3×C4×C8D411yescharacteristic
71C3⋊#95—21no—
72#830D412yes—
73#825—42no—
74(C2×D3)⋊C8—22no—
75Dic3⋊C8—22no—
76C8×Dic3—22no—
77#810—21no—
78#810—21no—
79#809—21no—
80#809—21no—
81#808—21no—
82#808—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C4×#41—31no—
92#479—61no—
93#478—61no—
94#477—31no—
95#476—31no—
96#475—31no—
97#474—31no—
98C8×Q8—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—31no—
102#431—61no—
103#427—61no—
104#423C2×D311yescharacteristic
105C2×C4⋊C8—31no—
106C2×#104—61no—
107#416—61no—
108C2×C4×C8—61no—
109C2×C4×C8C2×D311yescharacteristic
110C4×C4⋊C4—31no—
111#339—61no—
112#328—62no—
113Q8⋊C8—21no—
114Q8⋊C8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—21no—
121C22×Dic3—41no—
122#235—21no—
123#235—21no—
124#235SD1612yes—
125#235Q1612yes—
126C2×D12C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×Dic6C22⋊C411yescharacteristic
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154C4⋊Dic3C22⋊C411yescharacteristic
155C4⋊Dic3C22⋊C411yescharacteristic
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168D3×C8—44no—
169C2×#41—31no—
170C2×M16—21no—
171C22×C8—21no—
172C22×C8—62no—
173C22×C8—61no—
174C4⋊Q8C4×D311yescharacteristic
175C4⋊Q8C4×D311yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—31no—
182C4×Q8—31no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187#117—21no—
188#117—31no—
189#117—61no—
190C2×C4⋊C4—61no—
191C2×C42—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—61no—
195#107—61no—
196#106—61no—
197#106—61no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—62no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#104—122no—
206#103—61no—
207#103—61no—
208#102—61no—
209#102—61no—
210C22⋊C8—32no—
211C22⋊C8—62no—
212C4×C8—21no—
213C4×C8—62no—
214C4×C8—61no—
215C4×C8#6711yescharacteristic
216C4×C8#6711yescharacteristic
217#95—61no—
218C23×C3#12111yescharacteristic
219C22×D3—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—81no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×SD1611yescharacteristic
235C2×C3×C4C2×Q1611yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4#13511yescharacteristic
238#67—22no—
239#67—22no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—42no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246D12—21no—
247D12#10312yes—
248C4×D3—22no—
249C4×D3—22no—
250C4×D3—42no—
251C4×D3—41no—
252C4×D3—41no—
253Dic6—21no—
254Dic6#10312yes—
255C3×C8—22no—
256C3×C8—41no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261#41—32no—
262#41—32no—
263#41—61no—
264#41—61no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—31no—
268C2×D4—31no—
269C2×D4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4#23511yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—62no—
286C2×C8—124no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—31no—
294C4⋊C4—31no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—62no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—121no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C42—41no—
306C42—61no—
307C42—61no—
308C42—61no—
309C42—121no—
310C42—121no—
311C42C2×C4×D311yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#46511yescharacteristic
319C22×C3#46711yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4C2×#10311yescharacteristic
329C3×C4#41611yescharacteristic
330C3×C4#44311yescharacteristic
331C3×C4#44411yescharacteristic
332C3×C4#46511yescharacteristic
333C3×C4#46711yescharacteristic
334Dic3—22no—
335Dic3—41no—
336Dic3—41no—
337Dic3—82no—
338C23—61no—
339C23#88411yescharacteristic
340Q8—32no—
341Q8—41no—
342Q8—41no—
343Q8—61no—
344Q8—81no—
345D4—32no—
346D4—62no—
347D4—62no—
348D4—61no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—61no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4D3×C22⋊C411yescharacteristic
366C2×C4#67⋊C411yescharacteristic
367C2×C4#88311yescharacteristic
368C2×C4D3×SD1611yescharacteristic
369C2×C4#91611yescharacteristic
370C2×C4D3×Q1611yescharacteristic
371C2×C4#91911yescharacteristic
372C2×C4C4×#6711yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#93811yescharacteristic
375C8—22no—
376C8—41no—
377C8—81no—
378C8—122no—
379C8—121no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3#187611yescharacteristic
384C2×C3#189211yescharacteristic
385C2×C3#190611yescharacteristic
386D3—42no—
387C22—21no—
388C22—21no—
389C22—62no—
390C22—121no—
391C22—121no—
392C22#480711yescharacteristic
393C22#481311yescharacteristic
394C22#495611yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—62no—
399C4—81no—
400C4—81no—
401C4—121no—
402C4—121no—
403C4—242no—
404C4D3×#10311yescharacteristic
405C4#480411yescharacteristic
406C4#514411yescharacteristic
407C4#514511yescharacteristic
408C4#517111yescharacteristic
409C4#518811yescharacteristic
410C3#920711yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—122no—
414C2#7001111yescharacteristic
415C2#7027011yescharacteristic
416C2#7039111yescharacteristic
subgroupG/H
centerC2×C4#938
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#9207
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
3. C3×#1493C2semidirect
4. #71835C2semidirect
6. #67714C2semidirect
7. #67655C2semidirect
15. #5094C4semidirect
16. #5091C4semidirect
17. #5087C4semidirect
18. #5084C4semidirect
38. #1493D3semidirect
410. C3#9207semidirect

Maximal subgroups

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

subgroupindex
#833492
C8×#2352
C3×#14932
#718352
#705292
#677142
#676552
#92073