Finite Group Databasebeta

← #10691021

Subgroups — #10691021

Order 768.

486 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 95 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#83353C211yesmaximal, characteristic
2#79336C211yesmaximal, characteristic
3C3×#1531C211yesmaximal, characteristic, complemented
4#71835C211yesmaximal, characteristic, complemented
5#69812C211yesmaximal, characteristic
6#67749C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#9165—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#821—21no—
14#5087C411yescharacteristic, complemented
15#5087C411yescharacteristic, complemented
16#5083C411yescharacteristic, complemented
17#5083C411yescharacteristic, complemented
18#5046—21no—
19#5044—21no—
20#5026—21no—
21#5026—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25#4838—21no—
26#4834—21no—
27#4726—21no—
28D3×C22⋊C8—21no—
29#4706—22no—
30C3×#330—22no—
31#4551—21no—
32#4538—21no—
33#4538—21no—
34#3110—31no—
35#2974—31no—
36#1762—31no—
37#1531D311yescharacteristic, complemented
38#1503—31no—
39#1302—31no—
40#1285—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—21no—
54C2×#215—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4×Dic3—21no—
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12C2×C411yescharacteristic
64C4×D12C2×C411yescharacteristic
65C4×D12D411yescharacteristic
66C4×D12D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72#835—21no—
73#835—21no—
74C3⋊#95—21no—
75#830—21no—
76#830D411yescharacteristic
77#830D411yescharacteristic
78#821—24no—
79(C2×D3)⋊C8—22no—
80(C2×D3)⋊C8—22no—
81#809—21no—
82#809—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88#803—21no—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×C4×D4—31no—
92#484—61no—
93#474—31no—
94#474—31no—
95#461—31no—
96#461—31no—
97#450—61no—
98#440—61no—
99#440—61no—
100#436—61no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432C2×D311yescharacteristic
105#431—31no—
106C2×#102—61no—
107#407—61no—
108C2×C22⋊C8—61no—
109C2×#97C2×D311yescharacteristic
110C4×C4⋊C4—31no—
111#340—61no—
112#330—22no—
113#328—61no—
114#328—61no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—41no—
123C2×D12—22no—
124C2×D12—21no—
125C2×D12—41no—
126C2×D12C22⋊C411yescharacteristic
127C2×D12C22⋊C411yescharacteristic
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—82no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#215—42no—
150#215—42no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—41no—
158C4⋊Dic3C22⋊C411yescharacteristic
159C4⋊Dic3C22⋊C411yescharacteristic
160C4×Dic3—81no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165ΓL(1,𝔽25)—82no—
166D3×C8—82no—
167C22×D4—31no—
168C23×C4—61no—
169C2×SD16—61no—
170C2×SD16—61no—
171C2×M16—62no—
172C22×C8—21no—
173C22×C8—62no—
174C4⋊Q8C4×D311yescharacteristic
175C4⋊Q8C4×D311yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—31no—
182C4×D4—31no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187#117—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193#107—61no—
194#107—61no—
195C4⋊C8—21no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198C4⋊C8—61no—
199#103—61no—
200#103—61no—
201#102—61no—
202#102—61no—
203#102—64no—
204C22⋊C8—21no—
205C22⋊C8—31no—
206C22⋊C8—31no—
207C22⋊C8—62no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210#97—21no—
211#97—21no—
212#97—61no—
213#97#6712yes—
214C4×C8—61no—
215#95—61no—
216C23×C3#12111yescharacteristic
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—42no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—81no—
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—81no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—42no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245D12—21no—
246D12—21no—
247D12—21no—
248D12—42no—
249D12#10412yes—
250C4×D3—42no—
251C4×D3—42no—
252C4×D3—81no—
253C4×D3—81no—
254C3×C8—42no—
255C3×C8—42no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C24—61no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—31no—
264C2×D4—31no—
265C2×D4—62no—
266C2×D4—61no—
267C2×D4—121no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—242no—
275C22×C4#23511yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278SD16—122no—
279SD16—122no—
280M16—242no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—242no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—62no—
299C4⋊C4—61no—
300C4⋊C4—121no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C42—41no—
307C42—241no—
308C42C2×C4×D311yescharacteristic
309C42D3×D411yescharacteristic
310C42#23711yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#38611yescharacteristic
315C22×C3#47111yescharacteristic
316C22×C3#47311yescharacteristic
317C2×D3—22no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—82no—
324C2×D3—82no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—81no—
331C3×C4—81no—
332C3×C4C2×#10411yescharacteristic
333C3×C4#42111yescharacteristic
334Dic3—41no—
335Dic3—41no—
336Dic3—82no—
337C23—61no—
338C23—61no—
339C23—122no—
340C23—121no—
341C23—121no—
342C23—241no—
343C23#88411yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—81no—
347D4—32no—
348D4—61no—
349D4—61no—
350D4—61no—
351D4—122no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4—241no—
368C2×C4—241no—
369C2×C4D3×C22⋊C411yescharacteristic
370C2×C4#67⋊C411yescharacteristic
371C2×C4#88311yescharacteristic
372C2×C4C4×#6711yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#93811yescharacteristic
375C8—42no—
376C8—42no—
377C8—121no—
378C8—121no—
379C8—121no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3#187111yescharacteristic
384C2×C3#190411yescharacteristic
385C2×C3#190811yescharacteristic
386D3—42no—
387D3—81no—
388C22—21no—
389C22—21no—
390C22—62no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22—242no—
397C22—242no—
398C22#475411yescharacteristic
399C22#475511yescharacteristic
400C22#495611yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—81no—
407C4—81no—
408C4—121no—
409C4—121no—
410C4—242no—
411C4D3×#10411yescharacteristic
412C4#482111yescharacteristic
413C3#916511yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—122no—
417C2—241no—
418C2#7012311yescharacteristic
419C2#7038111yescharacteristic
420C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9165
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
3. C3×#1531C2semidirect
4. #71835C2semidirect
6. #67749C2semidirect
7. #67662C2semidirect
14. #5087C4semidirect
15. #5087C4semidirect
16. #5083C4semidirect
17. #5083C4semidirect
37. #1531D3semidirect
413. C3#9165semidirect

Maximal subgroups

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

subgroupindex
#833532
#793362
C3×#15312
#718352
#698122
#677492
#676622
#91653