Finite Group Databasebeta

← #10690380

Subgroups — #10690380

Order 768.

544 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2932C211yesmaximal, characteristic, complemented
2#83298C211yesmaximal, characteristic
3#71804C211yesmaximal, characteristic, complemented
4#70496C211yesmaximal, characteristic, complemented
5#69837C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67661C211yesmaximal, characteristic, complemented
8#9049—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×#532C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#435—22no—
13C3×#435—22no—
14C3×#423C2211yescharacteristic
15C3×#408—21no—
16C2×C3×#97C2211yescharacteristic
17C2×(C2×D3)⋊C8—21no—
18C2×(C2×D3)⋊C8—21no—
19#5087—21no—
20#5072—21no—
21#5036C412yes—
22#5033C412yes—
23C4×#213—22no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4547—21no—
28#4544—21no—
29#4483—22no—
30#3100—31no—
31#2932D311yescharacteristic, complemented
32#1955—31no—
33#1540—31no—
34#1486—31no—
35#1301—31no—
36#1281—31no—
37C2×C3×#41—21no—
38C22×D12C2×C411yescharacteristic
39C22×C4×D3—21no—
40C3×#131—42no—
41C3×#131—42no—
42C2×C3×M16—22no—
43C2×C3×M16—21no—
44C22×C3×C8—22no—
45C22×C3×C8D411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121C2×C412yes—
52C3×C4×D4—21no—
53C3×C4×D4D412yes—
54C2×C3×C42C2311yescharacteristic
55C2×#213—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C22×C3⋊C8—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D412yes—
65C3×C22⋊C8—24no—
66C3×#97—21no—
67C3×#97—21no—
68#832—22no—
69#830—21no—
70#830C2×C412yes—
71(C2×D3)⋊C8—42no—
72(C2×D3)⋊C8—42no—
73#809—22no—
74#807—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81C2×#131—21no—
82#532C2×D311yescharacteristic
83C2×C4×D4—31no—
84#480—32no—
85#474—61no—
86#459—61no—
87#447—32no—
88C4×D8—62no—
89#435—22no—
90#435—22no—
91#431—31no—
92#431—31no—
93#423C2×D311yescharacteristic
94C2×C4⋊C8—31no—
95#408—21no—
96C2×C22⋊C8—61no—
97C2×C22⋊C8—61no—
98C2×#97C2×D311yescharacteristic
99#332—61no—
100#331—61no—
101D4⋊C8—62no—
102C23×D3—21no—
103C3×#41—41no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C2×C3×Q8—21no—
108C2×C3×D4—22no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4C22⋊C412yes—
112C22×C3×C4—22no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—42no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C3×M16—44no—
127C3×M16—41no—
128C3×M16—41no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—22no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—44no—
136C2×C3×C8D812yescomplemented
137C3×C4⋊C4—21no—
138C3×C4⋊C4C22⋊C412yes—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#213—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×#41—21no—
156C22×D4—31no—
157C23×C4—61no—
158C2×D8—62no—
159#131—42no—
160#131—42no—
161C2×M16—22no—
162C2×M16—21no—
163C22×C8—22no—
164C22×C8—61no—
165C22×C8#6711yescharacteristic
166C22×C8#6711yescharacteristic
167C4⋊Q8D1211yescharacteristic
168#127D1211yescharacteristic
169#124—21no—
170#121—21no—
171#121C4×D312yes—
172C4×D4—21no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4#6712yes—
178C2×C4⋊C4—31no—
179C2×C22⋊C4—61no—
180C2×C42C22×D311yescharacteristic
181#107—62no—
182C4⋊C8—21no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8#6712yes—
188#102—62no—
189#102—62no—
190C22⋊C8—24no—
191C22⋊C8—32no—
192C22⋊C8—61no—
193C22⋊C8—122no—
194C22⋊C8—122no—
195#97—21no—
196#97—21no—
197C4×C8—61no—
198C4×C8—61no—
199C23×C3—22no—
200C23×C3C4×D411yescharacteristic
201C22×D3—22no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—81no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12311yescharacteristic
227C2×C3×C4C2×D811yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234D12—22no—
235D12—22no—
236D12—41no—
237D12—41no—
238C4×D3—81no—
239C4×D3—81no—
240C4×D3—81no—
241C4×D3—81no—
242C3×C8—22no—
243C3×C8—42no—
244C3×C8—41no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C3⋊C8—81no—
248C24—61no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—122no—
260C2×D4(C2×D3)⋊C412yes—
261C22×C4—22no—
262C22×C4—61no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C2×C4×D311yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4C2×#6711yescharacteristic
272D8—124no—
273M16—44no—
274M16—41no—
275M16—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—44no—
283C2×C8—62no—
284C2×C8—121no—
285C2×C8—121no—
286C2×C8—121no—
287C2×C8—121no—
288C2×C8#21312yescomplemented
289C4⋊C4—21no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4(C2×D3)⋊C412yes—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C42C2×D1211yescharacteristic
298C42#23511yescharacteristic
299C42C2×#6711yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3#38611yescharacteristic
307C22×C3C4×D811yescharacteristic
308C22×C3#44211yescharacteristic
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—42no—
321C3×C4#44711yescharacteristic
322C3×C4#44911yescharacteristic
323C3×C4#45911yescharacteristic
324C3×C4#46311yescharacteristic
325Dic3—81no—
326C23—22no—
327C23—62no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23C4×#6711yescharacteristic
335Q8—41no—
336Q8—41no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—41no—
341D4—41no—
342D4—62no—
343D4—62no—
344D4—81no—
345D4—121no—
346D4—121no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—241no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4C2×(C2×D3)⋊C411yescharacteristic
360C2×C4C4×#6711yescharacteristic
361C2×C4#92911yescharacteristic
362C2×C4#93011yescharacteristic
363C2×C4#93011yescharacteristic
364C2×C4C2×#21311yescharacteristic
365C2×C4#93211yescharacteristic
366C2×C4#94911yescharacteristic
367C2×C4#95011yescharacteristic
368C2×C4#95311yescharacteristic
369C8—22no—
370C8—42no—
371C8—41no—
372C8—122no—
373C8—241no—
374C8—241no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—82no—
378C2×C3#187911yescharacteristic
379C2×C3#188111yescharacteristic
380C2×C3#189111yescharacteristic
381D3—82no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—122no—
388C22—122no—
389C22—122no—
390C22—241no—
391C22—241no—
392C22—241no—
393C22—241no—
394C22C4×#21311yescharacteristic
395C22#501411yescharacteristic
396C22#521311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—241no—
403C4#501511yescharacteristic
404C4#501611yescharacteristic
405C4#521711yescharacteristic
406C4#523811yescharacteristic
407C3#904911yescharacteristic, complemented, Sylow
408C2—21no—
409C2—21no—
410C2—82no—
411C2—242no—
412C2#7161311yescharacteristic
413C2#7202011yescharacteristic
414C2#7202911yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2932C2
socleC22×C3#386
prime radical#83298C2
cosocleC2×C3×C42C23
hypercenter#2932D3
hypocenterC3#9049
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
1. C3×#2932C2semidirect
3. #71804C2semidirect
4. #70496C2semidirect
5. #69837C2semidirect
6. #67714C2semidirect
7. #67661C2semidirect
31. #2932D3semidirect
136. C2×C3×C8D8semidirect
288. C2×C8#213semidirect
407. C3#9049semidirect

Maximal subgroups

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

subgroupindex
C3×#29322
#832982
#718042
#704962
#698372
#677142
#676612
#90493