Finite Group Databasebeta

← #10695811

Subgroups — #10695811

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#82630C211yesmaximal, characteristic
2#82616C211yesmaximal, characteristic
3#78625C211yesmaximal, characteristic, complemented
4C3×#1471C211yesmaximal, characteristic
5#69923C211yesmaximal, characteristic, complemented
6#69911C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9978—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5218—21no—
15#5215—21no—
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5121—21no—
19#5120—21no—
20#5041—22no—
21#5040—21no—
22#5035—22no—
23#5034—21no—
24#5034—21no—
25#5033—21no—
26#4971—21no—
27#4964—21no—
28#4723—22no—
29#4722—21no—
30#4721—21no—
31C3×#323—22no—
32#4550—22no—
33#4469—22no—
34#3187—31no—
35#3095—31no—
36#3094—31no—
37#1818—31no—
38#1472—31no—
39#1471D311yescharacteristic
40#1293—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#214—22no—
56#926—21no—
57#926—21no—
58#926—21no—
59#926—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C22×C3⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75#832—22no—
76#832—22no—
77#832—22no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830D412yes—
82#818—22no—
83#817—22no—
84#816—22no—
85#816—22no—
86#810—22no—
87#808—22no—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×#121C2×D311yescharacteristic
92C2×#117—31no—
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#407C2×D311yescharacteristic
110C2×C22⋊C8—31no—
111C2×C22⋊C8C2×D311yescharacteristic
112#336—62no—
113#324—62no—
114#323—22no—
115C24×C3—21no—
116C24×C3#4111yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—42no—
138C3×M16—42no—
139C2×C3×C8—22no—
140C2×C3×C8—42no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146#217—41no—
147#217—41no—
148#217—41no—
149#217—41no—
150#214—82no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—42no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—42no—
165C22×Q8—31no—
166C22×D4—21no—
167C22×D4D1211yescharacteristic
168C23×C4C22×D311yescharacteristic
169C2×SD16—62no—
170C2×M16—21no—
171C2×M16—61no—
172C22×C8—21no—
173C22×C8—32no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—21no—
179#121—21no—
180#121—21no—
181#121—21no—
182#117—61no—
183#117—61no—
184#117—61no—
185#117—61no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4D1211yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—61no—
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—32no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8#6712yes—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12111yescharacteristic
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—82no—
227C3×D4—82no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—81no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—81no—
248Dic6—42no—
249Dic6—42no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C24—21no—
255C24#23511yescharacteristic
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—122no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—82no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—61no—
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—42no—
279M16—122no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—122no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—61no—
289C4⋊C4—61no—
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—121no—
303C42—121no—
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—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—162no—
317C22×C3#39211yescharacteristic
318C22×C3#47112yes—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—81no—
323C3×C4#45812yes—
324Dic3—81no—
325Dic3—81no—
326Dic3—81no—
327Dic3—81no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—41no—
333C23—81no—
334C23—81no—
335C23#88411yescharacteristic
336C23#88411yescharacteristic
337C23#93011yescharacteristic
338Q8—122no—
339Q8—122no—
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
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—161no—
372C2×C3#201711yescharacteristic
373C2×C3#201911yescharacteristic
374C2×C3#202211yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—162no—
388C22#474812yes—
389C22#496011yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—241no—
395C4—241no—
396C4—241no—
397C4—241no—
398C4#474212yes—
399C3#997811yescharacteristic, complemented, Sylow
400C2—22no—
401C2—41no—
402C2—41no—
403C2—161no—
404C2#7001311yescharacteristic
405C2#7003111yescharacteristic
406C2#7009111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1471C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#1471D3
hypocenterC3#9978
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
3. #78625C2semidirect
5. #69923C2semidirect
6. #69911C2semidirect
7. #67687C2semidirect
399. C3#9978semidirect

Maximal subgroups

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

subgroupindex
#826302
#826162
#786252
C3×#14712
#699232
#699112
#676872
#99783