Finite Group Databasebeta

← #10691136

Subgroups — #10691136

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2357C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#71597C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic, complemented
6#67631C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9181—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C3×#390—21no—
13C3×#388—22no—
14C3×#382C2211yescharacteristic
15C3×#381—21no—
16C3×C4×C4⋊C4—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19#5019C412yescomplemented
20#5016—22no—
21#5015C412yescomplemented
22#4966—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26D4⋊(C3⋊C8)—22no—
27#4451—21no—
28#4451—21no—
29#4451—22no—
30#3036—31no—
31#2357D311yescharacteristic, complemented
32#1517—31no—
33#1502—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×#235—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4C2×C412yes—
43C3×C4×D4D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C42—21no—
51C2×C3×C42—22no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54#932—42no—
55#932—42no—
56C2×#213—22no—
57#930—21no—
58#930—21no—
59C2×#208—21no—
60#875—21no—
61#874C2×C411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869C2×C411yescharacteristic
65C3×#95—22no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69C3×#95D412yes—
70#830—21no—
71#830—21no—
72#810—22no—
73#809—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C2×AGL(1,ℤ/8)—61no—
81#532—31no—
82C2×C4×D4C2×D311yescharacteristic
83#463—62no—
84#461—32no—
85#459—32no—
86#431—31no—
87#431—31no—
88#423—31no—
89#417—61no—
90#390—21no—
91#388—22no—
92#382C2×D311yescharacteristic
93#381—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×D311yescharacteristic
96C4×C22⋊C4—21no—
97#327—61no—
98#327—61no—
99#327—62no—
100D4⋊C8—62no—
101C24×C3—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—82no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122#235—41no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4C22⋊C412yes—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—44no—
136C3×C22⋊C4—82no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42—82no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#214—44no—
145#213—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150#208—41no—
151#208—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×#41—61no—
156C22×D4#6711yescharacteristic
157C23×C4—21no—
158AGL(1,ℤ/8)—122no—
159AGL(1,ℤ/8)—122no—
160C2×SD16—62no—
161C2×D8—62no—
162C2×M16—61no—
163C4⋊Q8—31no—
164#127—31no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4C4×D312yes—
173C4×D4D1212yes—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—32no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187#103—62no—
188#102—62no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191C4×C8—61no—
192C4×C8—61no—
193#95—22no—
194#95—21no—
195#95#6711yescharacteristic
196#95#6711yescharacteristic
197#95#6712yes—
198C23×C3—22no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3#12011yescharacteristic
204C22×D3—41no—
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—41no—
208C3×D4—41no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—82no—
223C2×C3×C4—82no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234#67—81no—
235#67—81no—
236C2×Dic3—41no—
237D12—41no—
238D12—41no—
239D12—81no—
240D12—81no—
241C4×D3—81no—
242C4×D3—81no—
243Dic6—41no—
244Dic6—41no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C3⋊C8—81no—
248C24—21no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—61no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—121no—
263C2×D4(C2×D3)⋊C412yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—82no—
271C22×C4—121no—
272C22×C4C2×#6711yescharacteristic
273C22×C4C2×#6711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275SD16—124no—
276D8—124no—
277M16—121no—
278M16—121no—
279C2×C8—62no—
280C2×C8—121no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4(C2×D3)⋊C412yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—44no—
292C22⋊C4—82no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C42—41no—
296C42—41no—
297C42—82no—
298C42C2×C4×D311yescharacteristic
299C42C2×D1211yescharacteristic
300C42#23511yescharacteristic
301C42#23511yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3#38611yescharacteristic
310C22×C3#44911yescharacteristic
311C22×C3#44911yescharacteristic
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—42no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4#41611yescharacteristic
324C3×C4#41811yescharacteristic
325C3×C4#48211yescharacteristic
326C3×C4#48211yescharacteristic
327Dic3—81no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—121no—
334C23#95311yescharacteristic
335Q8—121no—
336Q8—121no—
337D4—22no—
338D4—22no—
339D4—41no—
340D4—41no—
341D4—121no—
342D4—121no—
343D4—241no—
344D4—241no—
345D4—241no—
346D4—241no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—121no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4C2×(C2×D3)⋊C411yescharacteristic
366C2×C4C4×#6711yescharacteristic
367C2×C4C4×#6711yescharacteristic
368C2×C4#92911yescharacteristic
369C2×C4#93011yescharacteristic
370C2×C4#93011yescharacteristic
371C2×C4#94911yescharacteristic
372C2×C4#94911yescharacteristic
373C2×C4#95011yescharacteristic
374C2×C4#95011yescharacteristic
375C8—122no—
376C8—241no—
377C8—241no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—82no—
381C2×C3#187511yescharacteristic
382C2×C3#190411yescharacteristic
383C2×C3#190611yescharacteristic
384D3—161no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—82no—
391C22—82no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22#521311yescharacteristic
396C22#523811yescharacteristic
397C22#523811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—81no—
405C4—82no—
406C4—241no—
407C4#500111yescharacteristic
408C4#500211yescharacteristic
409C4#500311yescharacteristic
410C4#500311yescharacteristic
411C3#918111yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—82no—
415C2—481no—
416C2#7158611yescharacteristic
417C2#7164511yescharacteristic
418C2#7165111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2357C2
socleC22×C3#386
prime radical#82307C2
cosocleC2×C3×C42C23
hypercenter#2357D3
hypocenterC3#9181
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. C3×#2357C2semidirect
3. #71597C2semidirect
4. #69545C2semidirect
5. #69469C2semidirect
6. #67631C2semidirect
7. #67620C2semidirect
19. #5019C4semidirect
21. #5015C4semidirect
31. #2357D3semidirect
411. C3#9181semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#823072
#715972
#695452
#694692
#676312
#676202
#91813