Finite Group Databasebeta

← #10695810

Subgroups — #10695810

Order 768.

518 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 83 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#82220C211yesmaximal, characteristic, complemented
2#82217C211yesmaximal, characteristic
3#78612C211yesmaximal, characteristic
4C3×#1818C211yesmaximal, characteristic, complemented
5#69923C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic
7#67679C211yesmaximal, characteristic, complemented
8#9978—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5207C2211yescharacteristic
15#5207C2211yescharacteristic
16#5114—22no—
17#5004—21no—
18#5003—22no—
19#5001—21no—
20#4996—22no—
21#4995—21no—
22#4995—21no—
23#4971—21no—
24#4970—21no—
25#4968—21no—
26#4967—21no—
27#4738—21no—
28#4735—21no—
29#4731—21no—
30#4721—21no—
31C3×#336—22no—
32#4469—22no—
33#4468—22no—
34#3187—31no—
35#3095—31no—
36#3094—31no—
37#1818D311yescharacteristic, complemented
38#1472—31no—
39#1471—31no—
40#1293—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8D412yes—
47C3×#117—21no—
48C3×#117—21no—
49C3×#117—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×#208—21no—
69C2×#208—21no—
70C2×#204—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75#830—21no—
76#830—21no—
77#830D412yes—
78#830D412yes—
79#821—22no—
80#817—22no—
81#816—22no—
82#810—22no—
83#810—22no—
84#809—22no—
85#809—22no—
86#808—22no—
87#808—22no—
88#807—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×#121—31no—
92C2×#117C2×D311yescharacteristic
93#485—62no—
94#484—61no—
95#483—61no—
96#483—61no—
97#482—62no—
98#481—61no—
99#481—61no—
100#480—61no—
101#465—62no—
102#450—61no—
103#448—61no—
104#428—61no—
105#428—61no—
106#419—61no—
107#418—61no—
108#407—31no—
109#407—31no—
110C2×C22⋊C8C2×D311yescharacteristic
111C2×C22⋊C8C2×D311yescharacteristic
112#336—22no—
113#324—62no—
114#323—62no—
115C24×C3#4111yescharacteristic
116C23×D3—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—82no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—42no—
138C2×C3×C8—22no—
139C2×C3×C8—42no—
140C2×C3×C8—42no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150#217—41no—
151#217—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160#208—42no—
161#208—42no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164#204—82no—
165C22×Q8—31no—
166C22×D4—31no—
167C22×D4—61no—
168C23×C4C22×D311yescharacteristic
169C2×SD16—62no—
170C2×M16—61no—
171C2×M16—61no—
172C22×C8—21no—
173C22×C8#6712yes—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181#121—61no—
182#117—21no—
183#117—21no—
184#117—21no—
185#117—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4D1211yescharacteristic
190C2×C4⋊C4D1211yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—21no—
195#107—62no—
196#107—62no—
197#106—62no—
198#106—62no—
199#106—62no—
200#103—62no—
201#103—62no—
202#103—62no—
203#102—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3#12111yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3#12111yescharacteristic
220C22×D3—41no—
221C22×D3—42no—
222C22×D3—81no—
223C22×D3—81no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12711yescharacteristic
238#67—82no—
239#67—82no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246D12—42no—
247D12—42no—
248Dic6—42no—
249Dic6—42no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C24—61no—
255C24#23511yescharacteristic
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—122no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—122no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—242no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4C2×D1211yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4C2×#6711yescharacteristic
277SD16—242no—
278M16—122no—
279M16—122no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—41no—
303C42—41no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—82no—
313C22×C3#39211yescharacteristic
314C22×C3#47112yes—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—162no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4#45812yes—
325Dic3—81no—
326Dic3—81no—
327Dic3—81no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—122no—
332C23—121no—
333C23—241no—
334C23—241no—
335C23#88411yescharacteristic
336C23#88411yescharacteristic
337C23#93011yescharacteristic
338Q8—122no—
339Q8—122no—
340D4—122no—
341D4—122no—
342D4—242no—
343D4—242no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4#88211yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#94011yescharacteristic
364C8—42no—
365C8—81no—
366C8—122no—
367C8—122no—
368C2×C3—22no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3#201711yescharacteristic
372C2×C3#201911yescharacteristic
373C2×C3#202211yescharacteristic
374D3—161no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—82no—
384C22—241no—
385C22—241no—
386C22—241no—
387C22—482no—
388C22#496011yescharacteristic
389C22#514112yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—81no—
395C4—241no—
396C4—241no—
397C4—241no—
398C4#505312yes—
399C3#997811yescharacteristic, complemented, Sylow
400C2—22no—
401C2—41no—
402C2—41no—
403C2—481no—
404C2#6953911yescharacteristic
405C2#7000711yescharacteristic
406C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1818C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1818D3
hypocenterC3#9978
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
1. #82220C2semidirect
4. C3×#1818C2semidirect
5. #69923C2semidirect
7. #67679C2semidirect
37. #1818D3semidirect
399. C3#9978semidirect

Maximal subgroups

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

subgroupindex
#822202
#822172
#786122
C3×#18182
#699232
#699212
#676792
#99783