Finite Group Databasebeta

← #10691126

Subgroups — #10691126

Order 768.

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

normal subgroups only

This group has 544 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#83314C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic, complemented
3C3×#1517C211yesmaximal, characteristic
4#71811C211yesmaximal, characteristic
5#70767C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9180—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#887—21no—
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5232—21no—
16#5090C412yescomplemented
17#5075C412yescomplemented
18#5054—22no—
19#4972—21no—
20C4×C4⋊Dic3C2211yescharacteristic
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4650—21no—
24#4649—21no—
25#4647—22no—
26C3×#327—22no—
27#4537—21no—
28#4537—21no—
29#4456—22no—
30#3037—31no—
31#2351—31no—
32#1517D311yescharacteristic
33#1502—31no—
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C2×C3×#41—21no—
38C22×Dic6D411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#127C2×C411yescharacteristic
41C3×#124—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#951—42no—
49#951—42no—
50C2×#216—22no—
51C2×#214—22no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—22no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61#887—42no—
62#887—42no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6C2×C412yes—
66C4×Dic6D412yes—
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×C412yes—
81C4×C3⋊C8—21no—
82C2×#137—61no—
83#532C2×D311yescharacteristic
84C2×C4×Q8—31no—
85C2×C4×Q8—61no—
86#464—62no—
87#462—32no—
88#460—32no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92#417—61no—
93#389—61no—
94#389—62no—
95#382—61no—
96#382C2×D311yescharacteristic
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99#327—22no—
100#327—61no—
101#327—61no—
102Q8⋊C8—62no—
103C3×#41—41no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6C22⋊C412yes—
125C2×Dic6D812yes—
126C2×Dic6SD1612yes—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138#216—44no—
139#214—44no—
140C4⋊Dic3—21no—
141C4⋊Dic3—42no—
142C4⋊Dic3C22⋊C412yes—
143Dic3⋊C4—22no—
144Dic3⋊C4—22no—
145Dic3⋊C4—44no—
146Dic3⋊C4—44no—
147C4×Dic3—22no—
148C4×Dic3—22no—
149C4×Dic3—44no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×#41—21no—
157C22×Q8—31no—
158#137—122no—
159#137—122no—
160C2×Q16—62no—
161C2×SD16—62no—
162C2×M16—61no—
163C4⋊Q8C4×D311yescharacteristic
164#127C4×D311yescharacteristic
165#124—21no—
166#121—21no—
167#121—21no—
168C4×Q8—32no—
169C4×Q8—32no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—122no—
173C4×Q8—122no—
174C4×D4—21no—
175C4×D4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—61no—
182C2×C42—61no—
183C2×C42—62no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C4×C8—21no—
195C4×C8—61no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—62no—
200#95#6712yes—
201C23×C3—41no—
202C23×C3#12011yescharacteristic
203C3×Q8—41no—
204C3×Q8—41no—
205C3×D4—41no—
206C3×D4—41no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C3×D4—81no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×D811yescharacteristic
226C2×C3×C4C2×SD1611yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240Dic6—22no—
241Dic6—22no—
242Dic6—22no—
243Dic6#10214yes—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—32no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4C2×#6711yescharacteristic
271Q16—124no—
272SD16—124no—
273M16—121no—
274M16—121no—
275C2×C8—41no—
276C2×C8—62no—
277C2×C8—121no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—32no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—124no—
287C4⋊C4—124no—
288C4⋊C4—122no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—62no—
292C42—62no—
293C42—124no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42D3×D411yescharacteristic
299C42#23711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3#38611yescharacteristic
306C22×C3#44711yescharacteristic
307C22×C3#44811yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313C3×C4—81no—
314C3×C4C2×#10211yescharacteristic
315C3×C4#41711yescharacteristic
316C3×C4#48011yescharacteristic
317C3×C4#48111yescharacteristic
318Dic3—24no—
319Dic3—42no—
320Dic3—42no—
321Dic3—81no—
322C23—41no—
323C23#93711yescharacteristic
324Q8—34no—
325Q8—41no—
326Q8—41no—
327Q8—62no—
328Q8—62no—
329Q8—62no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4D3×C22⋊C411yescharacteristic
356C2×C4#67⋊C411yescharacteristic
357C2×C4#88311yescharacteristic
358C2×C4#88411yescharacteristic
359C2×C4D3×D811yescharacteristic
360C2×C4#91111yescharacteristic
361C2×C4D3×SD1611yescharacteristic
362C2×C4#91511yescharacteristic
363C2×C4C4×#6711yescharacteristic
364C2×C4#93811yescharacteristic
365C8—81no—
366C8—122no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—161no—
371C2×C3#187411yescharacteristic
372C2×C3#190511yescharacteristic
373C2×C3#190611yescharacteristic
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#495611yescharacteristic
380C22#515611yescharacteristic
381C22#517011yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—64no—
387C4—82no—
388C4—81no—
389C4—122no—
390C4—122no—
391C4—241no—
392C4D3×#10211yescharacteristic
393C4#477111yescharacteristic
394C4#477411yescharacteristic
395C4#477711yescharacteristic
396C3#918011yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—161no—
400C2#7005511yescharacteristic
401C2#7020011yescharacteristic
402C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1517D3
hypocenterC3#9180
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. #74157C2semidirect
5. #70767C2semidirect
6. #67746C2semidirect
7. #67631C2semidirect
16. #5090C4semidirect
17. #5075C4semidirect
396. C3#9180semidirect

Maximal subgroups

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

subgroupindex
#833142
#741572
C3×#15172
#718112
#707672
#677462
#676312
#91803