Finite Group Databasebeta

← #10691010

Subgroups — #10691010

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2974C211yesmaximal, characteristic, complemented
2#82310C211yesmaximal, characteristic
3#71609C211yesmaximal, characteristic, complemented
4#70806C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#67749C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#9163—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#436—21no—
12C3×#435—22no—
13C3×C8×D4—21no—
14C3×#432C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#407—21no—
17C2×C3×C22⋊C8—21no—
18#5217—21no—
19C2×#832—21no—
20#5015C411yescharacteristic, complemented
21#5015C411yescharacteristic, complemented
22#5014—21no—
23#5014—21no—
24#5008C411yescharacteristic, complemented
25#5008C411yescharacteristic, complemented
26#4968—21no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4926C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4529—21no—
31#4470—21no—
32#4470—21no—
33#4464—22no—
34#3109—31no—
35#2974D311yescharacteristic, complemented
36#1762—31no—
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×M16—22no—
45C22×C3×C8—22no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4C2×C411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C3×C4×D4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#213—21no—
56C2×#213—21no—
57#930—21no—
58#930—21no—
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—22no—
70C3×C22⋊C8—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D411yescharacteristic
73C3×C22⋊C8D411yescharacteristic
74C3×#97D412yes—
75C3×C4×C8—21no—
76#832—24no—
77#832—21no—
78#832—21no—
79C3⋊#95—21no—
80#830—21no—
81#809—21no—
82#809—21no—
83#808—21no—
84#808—21no—
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87#803—21no—
88#803—21no—
89#803—21no—
90#532—31no—
91C2×C4×D4C2×D311yescharacteristic
92#481—61no—
93#476—31no—
94#476—31no—
95#459—31no—
96#459—31no—
97#447—61no—
98#442—61no—
99#442—61no—
100#436—21no—
101#435—22no—
102C8×D4—21no—
103#432—31no—
104#432C2×D311yescharacteristic
105#431C2×D311yescharacteristic
106C2×#102—61no—
107#407—21no—
108C2×C22⋊C8—21no—
109C2×#97—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#329—62no—
113#327—61no—
114#327—61no—
115C24×C3—21no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4C22⋊C411yescharacteristic
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—82no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×#67—41no—
131C22×Dic3—41no—
132#235—81no—
133#235—81no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C3×M16—82no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—22no—
143C2×C3×C8—82no—
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×C22⋊C4—21no—
151C3×C42C22×C411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#213—42no—
156#213—42no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—22no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—41no—
163C4×Dic3—81no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×#41—61no—
170C22×D4D1211yescharacteristic
171C23×C4—21no—
172C2×D8—61no—
173C2×D8—61no—
174C2×M16—22no—
175C22×C8—22no—
176C22×C8—61no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4C4×D311yescharacteristic
187C4×D4C4×D311yescharacteristic
188C4×D4#6711yescharacteristic
189C4×D4#6711yescharacteristic
190C2×C4⋊C4—61no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—61no—
196#106—61no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—31no—
200C4⋊C8—31no—
201#102—64no—
202#102—61no—
203#102—61no—
204#102—61no—
205#102—61no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8#6711yescharacteristic
211C22⋊C8#6711yescharacteristic
212#97—61no—
213#97—61no—
214#97—61no—
215#97#6712yes—
216C4×C8—21no—
217#95—61no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3#12111yescharacteristic
225C22×D3—41no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—42no—
230C3×D4#10412yes—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249#67—81no—
250#67—81no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255D12—41no—
256D12—41no—
257D12—81no—
258D12—81no—
259C4×D3—82no—
260Dic6—81no—
261C3×C8—41no—
262C3×C8—42no—
263C3×C8—41no—
264C3⋊C8—42no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C24—21no—
268#41—241no—
269#41—241no—
270C2×Q8—61no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—121no—
278C2×D4(C2×D3)⋊C411yescharacteristic
279C2×D4(C2×D3)⋊C411yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—82no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×D1211yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4C2×#6711yescharacteristic
291D8—122no—
292D8—122no—
293M16—82no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—82no—
301C2×C8—121no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—62no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C42—241no—
317C42C2×C4×D311yescharacteristic
318C42#23511yescharacteristic
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3#38611yescharacteristic
332C22×C3#47111yescharacteristic
333C22×C3#47211yescharacteristic
334C2×D3—81no—
335C2×D3—81no—
336C2×D3—81no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4C2×#10411yescharacteristic
345C3×C4#42111yescharacteristic
346Dic3—82no—
347Dic3—81no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—81no—
354C23—121no—
355C23#93011yescharacteristic
356Q8—241no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—121no—
362D4—121no—
363D4—241no—
364D4—241no—
365D4—241no—
366D4—241no—
367D4#82512yes—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—242no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4C2×(C2×D3)⋊C411yescharacteristic
386C2×C4C4×#6711yescharacteristic
387C2×C4C4×#6711yescharacteristic
388C2×C4#92911yescharacteristic
389C2×C4#93011yescharacteristic
390C2×C4#95311yescharacteristic
391C8—41no—
392C8—42no—
393C8—41no—
394C8—122no—
395C8—121no—
396C8—121no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—42no—
400C2×C3—81no—
401C2×C3#187111yescharacteristic
402C2×C3#190411yescharacteristic
403C2×C3#190711yescharacteristic
404D3—161no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22#514111yescharacteristic
419C22#514211yescharacteristic
420C22#521311yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—242no—
429C4—241no—
430C4C2×#82511yescharacteristic
431C4#513911yescharacteristic
432C3#916311yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—42no—
436C2—81no—
437C2—481no—
438C2#7165111yescharacteristic
439C2#7198411yescharacteristic
440C2#7201811yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2974C2
socleC22×C3#386
prime radical#82310C2
cosocleC2×C3×C42C23
hypercenter#2974D3
hypocenterC3#9163
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. C3×#2974C2semidirect
3. #71609C2semidirect
4. #70806C2semidirect
5. #70789C2semidirect
6. #67749C2semidirect
7. #67742C2semidirect
20. #5015C4semidirect
21. #5015C4semidirect
24. #5008C4semidirect
25. #5008C4semidirect
35. #2974D3semidirect
432. C3#9163semidirect

Maximal subgroups

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

subgroupindex
C3×#29742
#823102
#716092
#708062
#707892
#677492
#677422
#91633