Finite Group Databasebeta

← #10691119

Subgroups — #10691119

Order 768.

620 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83297C211yesmaximal, characteristic
2#74228C211yesmaximal, characteristic, complemented
3C3×#1517C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#70757C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9179—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#949—21no—
11C2×#893—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5232—21no—
16#5083C412yescomplemented
17#5072C412yescomplemented
18#5057—22no—
19#4972—21no—
20C4×C4⋊Dic3C2211yescharacteristic
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4668—22no—
24#4667—21no—
25D3×#95—21no—
26C3×#327—22no—
27#4537—21no—
28#4537—21no—
29#4459—22no—
30#3036—31no—
31#2349—31no—
32#1517D311yescharacteristic, complemented
33#1502—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C2×C3×#41—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×C4×D3—22no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#127C2×C411yescharacteristic, complemented
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#949—42no—
51#949—42no—
52C2×#215—22no—
53C2×#213—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61#893—42no—
62#893—42no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12C2×C412yes—
66C4×D12D412yes—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70C3×#95—21no—
71C3×#95D412yes—
72#835—22no—
73#832—22no—
74C3⋊#95—22no—
75C3⋊#95D411yescharacteristic
76C3⋊#95D411yescharacteristic
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yescomplemented
81C4×C3⋊C8—21no—
82C2×AGL(1,ℤ/8)—61no—
83#532C2×D311yescharacteristic
84C2×C4×D4—31no—
85C2×C4×D4—61no—
86#463—62no—
87#461—32no—
88#459—32no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92#417—61no—
93#389—61no—
94#385—62no—
95#382C2×D311yescharacteristic
96#380—61no—
97C4×C4⋊C4—31no—
98C2×#95—61no—
99#327—22no—
100#327—61no—
101#327—61no—
102D4⋊C8—62no—
103C23×D3—21no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12C22⋊C412yes—
125C2×D12D812yes—
126C2×D12SD1612yes—
127C2×C4×D3—22no—
128C2×C4×D3—22no—
129C2×C4×D3—44no—
130C2×C4×D3—44no—
131C2×C4×D3—82no—
132C2×C4×D3—82no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#215—44no—
145#213—44no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—44no—
149C4⋊Dic3—21no—
150C4⋊Dic3—42no—
151C4⋊Dic3C22⋊C412yes—
152Dic3⋊C4—82no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×#41—21no—
160C22×D4—31no—
161C23×C4—62no—
162C23×C4—61no—
163AGL(1,ℤ/8)—122no—
164AGL(1,ℤ/8)—122no—
165C2×SD16—62no—
166C2×D8—62no—
167C2×M16—61no—
168C4⋊Q8C4×D311yescharacteristic, complemented
169#127C4×D311yescharacteristic, complemented
170#124—21no—
171#121—21no—
172#121—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—32no—
176C4×D4—32no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—122no—
180C4×D4—122no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—62no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197C4×C8—21no—
198C4×C8—61no—
199#95—21no—
200#95—31no—
201#95—31no—
202#95—62no—
203#95#6712yes—
204C23×C3—41no—
205C23×C3#12011yescharacteristic
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—81no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4C2×SD1611yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245D12—22no—
246D12—22no—
247D12—22no—
248D12#10214yes—
249C4×D3—82no—
250C4×D3—82no—
251C4×D3—88no—
252C3×C8—81no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—61no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—32no—
265C2×D4—32no—
266C2×D4—32no—
267C2×D4—41no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4—124no—
279C22×C4—124no—
280C22×C4—242no—
281C22×C4—242no—
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4C2×#6711yescharacteristic
285SD16—124no—
286D8—124no—
287M16—121no—
288M16—121no—
289C2×C8—41no—
290C2×C8—62no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—32no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—61no—
298C4⋊C4—122no—
299C4⋊C4—242no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—124no—
305C42—121no—
306C42—121no—
307C42C2×C4×D311yescharacteristic
308C42#23511yescharacteristic
309C42D3×D411yescharacteristic
310C42#23711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#38611yescharacteristic
317C22×C3#44711yescharacteristic
318C22×C3#44811yescharacteristic
319C2×D3—24no—
320C2×D3—44no—
321C2×D3—44no—
322C2×D3—42no—
323C2×D3—42no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—82no—
329C3×C4—81no—
330C3×C4C2×#10211yescharacteristic
331C3×C4#41711yescharacteristic
332C3×C4#48011yescharacteristic
333C3×C4#48111yescharacteristic
334Dic3—42no—
335Dic3—42no—
336Dic3—81no—
337C23—41no—
338C23—62no—
339C23—62no—
340C23—62no—
341C23—122no—
342C23—122no—
343C23#93711yescharacteristic
344Q8—41no—
345Q8—41no—
346D4—34no—
347D4—41no—
348D4—41no—
349D4—62no—
350D4—62no—
351D4—62no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—242no—
373C2×C4—242no—
374C2×C4—248no—
375C2×C4D3×C22⋊C411yescharacteristic
376C2×C4#67⋊C411yescharacteristic
377C2×C4#88311yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4D3×D811yescharacteristic
380C2×C4#91111yescharacteristic
381C2×C4D3×SD1611yescharacteristic
382C2×C4#91511yescharacteristic
383C2×C4C4×#6711yescharacteristic
384C2×C4#93811yescharacteristic
385C8—81no—
386C8—122no—
387C8—241no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—161no—
391C2×C3#187411yescharacteristic
392C2×C3#190411yescharacteristic
393C2×C3#190611yescharacteristic
394D3—44no—
395C22—21no—
396C22—21no—
397C22—64no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—124no—
402C22—124no—
403C22—122no—
404C22—122no—
405C22#495611yescharacteristic
406C22#515611yescharacteristic
407C22#517011yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—82no—
413C4—81no—
414C4—122no—
415C4—122no—
416C4—241no—
417C4D3×#10211yescharacteristic
418C4#477111yescharacteristic
419C4#477411yescharacteristic
420C4#477711yescharacteristic
421C3#917911yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—124no—
425C2—161no—
426C2#7005511yescharacteristic
427C2#7020011yescharacteristic
428C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#386
prime radical#83297C2
cosocleC2×C3×C42C23
hypercenter#1517D3
hypocenterC3#9179
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #74228C2semidirect
3. C3×#1517C2semidirect
4. #71811C2semidirect
5. #70757C2semidirect
6. #67746C2semidirect
7. #67631C2semidirect
16. #5083C4semidirect
17. #5072C4semidirect
32. #1517D3semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×#127C2×C4semidirect
80. C4⋊(C3⋊C8)C2×C4semidirect
168. C4⋊Q8C4×D3semidirect
169. #127C4×D3semidirect
421. C3#9179semidirect

Maximal subgroups

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

subgroupindex
#832972
#742282
C3×#15172
#718112
#707572
#677462
#676312
#91793