Finite Group Databasebeta

← #10691998

Subgroups — #10691998

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2875C211yesmaximal, characteristic, complemented
2#82531C211yesmaximal, characteristic
3#71640C211yesmaximal, characteristic, complemented
4#69853C211yesmaximal, characteristic, complemented
5#69798C211yesmaximal, characteristic, complemented
6#67670C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#9311—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×C4×#41C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#405—21no—
13C3×#405—22no—
14C3×C4×M16—21no—
15C3×C4×M16C2211yescharacteristic
16C2×C3×#97—21no—
17C2×C3×#97C2211yescharacteristic
18#5238—21no—
19#5027C411yescharacteristic
20#5026—21no—
21#5020C411yescharacteristic
22#5015C411yescharacteristic
23#5014—21no—
24#5007C411yescharacteristic
25#5003—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31#4464—21no—
32#4464—21no—
33#4459—22no—
34#3115—31no—
35#2875D311yescharacteristic, complemented
36#1530—31no—
37#1488—31no—
38#1484—31no—
39#1302—31no—
40#1291—31no—
41C2×C3×#41D411yescharacteristic
42C2×#235—21no—
43C3×#131—24no—
44C3×#131—24no—
45C2×C3×M16—22no—
46C2×C3×M16—21no—
47C22×C3×C8—22no—
48C22×C3×C8—21no—
49C3×C4×Q8C2×C411yescharacteristic
50C3×C4×Q8D411yescharacteristic
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4C2×C411yescharacteristic
54C3×C4×D4D411yescharacteristic
55C3×#117—21no—
56C3×#117D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#215—21no—
60C2×#213—21no—
61#930—21no—
62#930—21no—
63#875—21no—
64#874C2×C411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic
68C3×#97—22no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D411yescharacteristic
72C3×#97D411yescharacteristic
73C3×C4×C8—22no—
74C3×C4×C8D412yes—
75#835—21no—
76#832—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#809—21no—
81#809—21no—
82#808—21no—
83#807—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88#803—21no—
89#803—21no—
90C4×C3⋊C8—21no—
91C2×#131—21no—
92#532—31no—
93C4×#41C2×D311yescharacteristic
94#482—61no—
95#475—31no—
96#474—31no—
97#460—31no—
98#459—31no—
99#449—61no—
100#442—61no—
101#440—61no—
102#432—31no—
103#432—31no—
104#431—31no—
105#405—21no—
106#405—22no—
107C4×M16—21no—
108C4×M16C2×D311yescharacteristic
109C2×#97—21no—
110C2×#97C2×D311yescharacteristic
111#329—61no—
112#329—61no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—62no—
116C3×#41—22no—
117C3×#41—22no—
118C2×C3×Q8C22⋊C411yescharacteristic
119C2×C3×D4—21no—
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128#235—82no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C3×M16—28no—
135C3×M16—22no—
136C3×M16—22no—
137C2×C3×C8—28no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—21no—
141C2×C3×C8—22no—
142C2×C3×C8—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#215—42no—
158#213—42no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×#41—61no—
168C2×#41#6711yescharacteristic
169C2×SD16—61no—
170C2×D8—61no—
171#131—24no—
172#131—24no—
173C2×M16—22no—
174C2×M16—21no—
175C22×C8—22no—
176C22×C8—21no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×Q8C4×D311yescharacteristic
183C4×Q8D1211yescharacteristic
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4C4×D311yescharacteristic
189C4×D4D1211yescharacteristic
190#117—21no—
191#117#6711yescharacteristic
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194#107—61no—
195#106—61no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—61no—
201#102—61no—
202#102—61no—
203#102—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207#97—22no—
208#97—21no—
209#97—21no—
210#97—61no—
211#97—61no—
212#97#6711yescharacteristic
213#97#6711yescharacteristic
214C4×C8—22no—
215C4×C8—61no—
216C4×C8#6712yes—
217C23×C3—21no—
218C23×C3#12311yescharacteristic
219C22×D3—41no—
220C3×Q8—21no—
221C3×Q8#10412yes—
222C3×D4—24no—
223C3×D4—21no—
224C3×D4#10412yes—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241#67—82no—
242C2×Dic3—41no—
243D12—42no—
244D12—41no—
245D12—41no—
246D12—81no—
247C4×D3—82no—
248Dic6—81no—
249C3×C8—22no—
250C3×C8—22no—
251C3×C8—24no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C3⋊C8—81no—
255C3⋊C8—81no—
256#41—22no—
257#41—22no—
258#41—242no—
259C2×Q8—61no—
260C2×Q8(C2×D3)⋊C411yescharacteristic
261C2×D4—21no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—121no—
266C2×D4(C2×D3)⋊C411yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—121no—
271C22×C4#23511yescharacteristic
272C22×C4#23511yescharacteristic
273C22×C4C2×#6711yescharacteristic
274SD16—122no—
275D8—122no—
276M16—28no—
277M16—22no—
278M16—22no—
279C2×C8—28no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4(C2×D3)⋊C411yescharacteristic
294C4⋊C4(C2×D3)⋊C411yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42C2×C4×D311yescharacteristic
304C42C2×D1211yescharacteristic
305C42C2×#6711yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3#38611yescharacteristic
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4C2×#10411yescharacteristic
326C3×C4C2×#10411yescharacteristic
327Dic3—81no—
328C23—21no—
329C23—121no—
330C23#92911yescharacteristic
331Q8—21no—
332Q8—241no—
333Q8#80512yes—
334D4—24no—
335D4—21no—
336D4—122no—
337D4—121no—
338D4—121no—
339D4—241no—
340D4—242no—
341D4#80512yes—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—121no—
353C2×C4—242no—
354C2×C4C2×(C2×D3)⋊C411yescharacteristic
355C2×C4C4×#6711yescharacteristic
356C2×C4C4×#6711yescharacteristic
357C2×C4#93011yescharacteristic
358C2×C4#93011yescharacteristic
359C2×C4#95311yescharacteristic
360C8—22no—
361C8—22no—
362C8—24no—
363C8—121no—
364C8—121no—
365C8—241no—
366C8—241no—
367C2×C3—41no—
368C2×C3—42no—
369C2×C3#188111yescharacteristic
370C2×C3#189611yescharacteristic
371C2×C3#190311yescharacteristic
372D3—161no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22#521311yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—241no—
392C4C2×#80511yescharacteristic
393C4C2×#80511yescharacteristic
394C3#931111yescharacteristic, complemented, Sylow
395C2—41no—
396C2—42no—
397C2—481no—
398C2#7153211yescharacteristic
399C2#7202011yescharacteristic
400C2#7202311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2875C2
socleC22×C3#386
prime radical#82531C2
cosocleC2×C3×C42C23
hypercenter#2875D3
hypocenterC3#9311
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2875

Complements

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

NH ≅ G/Ntype
1. C3×#2875C2semidirect
3. #71640C2semidirect
4. #69853C2semidirect
5. #69798C2semidirect
6. #67670C2semidirect
7. #67662C2semidirect
35. #2875D3semidirect
394. C3#9311semidirect

Maximal subgroups

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

subgroupindex
C3×#28752
#825312
#716402
#698532
#697982
#676702
#676622
#93113