Finite Group Databasebeta

← #10691312

Subgroups — #10691312

Order 768.

450 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2922C211yesmaximal, characteristic, complemented
2#82529C211yesmaximal, characteristic
3#71641C211yesmaximal, characteristic, complemented
4#69859C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic, complemented
6#67669C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9213—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#433—22no—
12C3×#431—22no—
13C3×#424—21no—
14C3×#408—21no—
15C3×C4×M16C2211yescharacteristic
16C2×C3×C4×C8C2211yescharacteristic
17#5238—21no—
18#5027C411yescharacteristic
19C4×#215—21no—
20#5021C411yescharacteristic
21#5015C411yescharacteristic
22C4×#213—21no—
23#5008C411yescharacteristic
24#5003—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28#4535—21no—
29D4⋊(C3⋊C8)—21no—
30#4464—22no—
31#4459—22no—
32#3106—31no—
33#2922D311yescharacteristic, complemented
34#1525—31no—
35#1492—31no—
36#1489—31no—
37#1299—31no—
38#1278—31no—
39C2×C3×#41D411yescharacteristic
40C2×#235—21no—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C3×C4×Q8C2×C411yescharacteristic
44C3×C4×Q8D411yescharacteristic
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4D411yescharacteristic
49C3×#117—21no—
50C3×#117D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—21no—
54C2×#213—21no—
55#930—21no—
56#930—21no—
57C2×#208—21no—
58C2×#208—21no—
59#875—21no—
60#874C2×C411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869C2×C411yescharacteristic
64C3×C4⋊C8—24no—
65C3×C22⋊C8—24no—
66C3×#97D412yes—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8D412yes—
70#835—21no—
71#832—21no—
72#830—21no—
73#809—21no—
74#809—21no—
75#808—21no—
76#807—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C411yescharacteristic
80C4⋊(C3⋊C8)C2×C411yescharacteristic
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#532—31no—
85C4×#41C2×D311yescharacteristic
86#482—61no—
87#477—31no—
88#476—31no—
89#460—31no—
90#459—31no—
91#449—61no—
92C4×SD16—61no—
93C4×D8—61no—
94#433—22no—
95#431—22no—
96#431—31no—
97#424—21no—
98#423—31no—
99#408—21no—
100C4×M16—31no—
101C4×M16C2×D311yescharacteristic
102C2×C4×C8C2×D311yescharacteristic
103#329—62no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106D4⋊C8—62no—
107C3×#41—41no—
108C3×#41—41no—
109C2×C3×Q8C22⋊C411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119#235—81no—
120#235—81no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×C4×D3—41no—
125C2×Dic6—21no—
126C3×M16—41no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4C22⋊C411yescharacteristic
135C3×C4⋊C4C22⋊C411yescharacteristic
136C3×C22⋊C4—21no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—21no—
139C3×C42—21no—
140C3×C42—21no—
141C3×C42—21no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146#215—42no—
147#213—42no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152#208—22no—
153#208—41no—
154#208—81no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×#41—61no—
160C2×#41#6711yescharacteristic
161C2×SD16—61no—
162C2×D8—61no—
163C2×M16—21no—
164C2×M16—61no—
165C2×M16—61no—
166C22×C8—21no—
167C4⋊Q8—31no—
168#127—31no—
169#124—61no—
170#121—61no—
171#121—61no—
172C4×Q8C4×D311yescharacteristic
173C4×Q8D1211yescharacteristic
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4C4×D311yescharacteristic
179C4×D4D1211yescharacteristic
180#117—21no—
181#117#6711yescharacteristic
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—61no—
185#106—61no—
186C4⋊C8—24no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—61no—
192#102—61no—
193#102—61no—
194#102—61no—
195C22⋊C8—24no—
196C22⋊C8—61no—
197#97—61no—
198#97#6712yes—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—61no—
202C4×C8—61no—
203C4×C8#6712yes—
204C23×C3—21no—
205C23×C3#12311yescharacteristic
206C22×D3—41no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×D4—21no—
210C3×D4—21no—
211C3×D4—42no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232#67—82no—
233C2×Dic3—41no—
234D12—41no—
235D12—41no—
236D12—41no—
237D12—41no—
238D12—81no—
239C4×D3—81no—
240C4×D3—81no—
241Dic6—81no—
242C3×C8—42no—
243C3×C8—42no—
244C3⋊C8—42no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—81no—
248#41—41no—
249#41—41no—
250#41—241no—
251#41—241no—
252C2×Q8—61no—
253C2×Q8(C2×D3)⋊C411yescharacteristic
254C2×D4—21no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4(C2×D3)⋊C411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—121no—
264C22×C4#23511yescharacteristic
265C22×C4#23511yescharacteristic
266C22×C4C2×#6711yescharacteristic
267SD16—122no—
268D8—122no—
269M16—41no—
270M16—41no—
271M16—62no—
272M16—121no—
273M16—241no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—121no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4(C2×D3)⋊C411yescharacteristic
287C4⋊C4(C2×D3)⋊C411yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C2×C4×D311yescharacteristic
297C42C2×D1211yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3#38611yescharacteristic
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4#42111yescharacteristic
317C3×C4#42111yescharacteristic
318C3×C4#44311yescharacteristic
319C3×C4#44411yescharacteristic
320C3×C4#47111yescharacteristic
321C3×C4#47211yescharacteristic
322Dic3—81no—
323C23—21no—
324C23—121no—
325C23#92911yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—241no—
329D4—21no—
330D4—21no—
331D4—42no—
332D4—121no—
333D4—121no—
334D4—121no—
335D4—121no—
336D4—241no—
337D4—242no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—121no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4C2×(C2×D3)⋊C411yescharacteristic
356C2×C4C4×#6711yescharacteristic
357C2×C4C4×#6711yescharacteristic
358C2×C4#93011yescharacteristic
359C2×C4#93011yescharacteristic
360C2×C4#95311yescharacteristic
361C8—42no—
362C8—42no—
363C8—122no—
364C8—121no—
365C8—121no—
366C8—241no—
367C2×C3—41no—
368C2×C3—81no—
369C2×C3#187611yescharacteristic
370C2×C3#189711yescharacteristic
371C2×C3#190711yescharacteristic
372D3—161no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22#521311yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—241no—
390C4#500511yescharacteristic
391C4#500511yescharacteristic
392C4#514111yescharacteristic
393C4#514211yescharacteristic
394C4#514411yescharacteristic
395C4#514511yescharacteristic
396C3#921311yescharacteristic, complemented, Sylow
397C2—41no—
398C2—81no—
399C2—481no—
400C2#7159111yescharacteristic
401C2#7201011yescharacteristic
402C2#7201811yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2922C2
socleC22×C3#386
prime radical#82529C2
cosocleC2×C3×C42C23
hypercenter#2922D3
hypocenterC3#9213
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
3. #71641C2semidirect
4. #69859C2semidirect
5. #69735C2semidirect
6. #67669C2semidirect
7. #67656C2semidirect
33. #2922D3semidirect
396. C3#9213semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
#825292
#716412
#698592
#697352
#676692
#676562
#92133