Finite Group Databasebeta

← #10697234

Subgroups — #10697234

Order 768.

518 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×C3×#475C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#82441C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71624C211yesmaximal, characteristic, complemented
6#70426C211yesmaximal, characteristic
7#67707C211yesmaximal, characteristic, complemented
8#10189—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#942—21no—
12#5477C2211yescharacteristic
13C3×#475—22no—
14C3×#475—22no—
15C2×C3×#106—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18#5088—21no—
19#5084—22no—
20#5080—21no—
21#5031—21no—
22#5028—22no—
23#5027—21no—
24#4968—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4528—22no—
31#4483—22no—
32#3227—31no—
33C2×#475D311yescharacteristic, complemented
34#3037—31no—
35#1548—31no—
36#1480—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C2×#235—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#216—22no—
58#942—42no—
59#942—42no—
60C2×#215—22no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×#106—24no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#103—24no—
73#830—21no—
74#830D412yes—
75#810—22no—
76#809—22no—
77#809—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8C2×D311yescharacteristic
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#475—22no—
94#475—22no—
95#464—62no—
96#462—61no—
97#460—61no—
98#431—31no—
99#431—31no—
100C2×#106—21no—
101#423—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103#417—61no—
104C2×#103—21no—
105#333—62no—
106D4⋊C8—62no—
107#324—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—82no—
116C2×C3×Q8SD1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×#67—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150#216—44no—
151#215—44no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156#208—41no—
157#208—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×#41—61no—
162C22×Q8—21no—
163C22×Q8#6711yescharacteristic
164#137—122no—
165#137—122no—
166C2×Q16—62no—
167C2×SD16—62no—
168C2×M16—61no—
169C22×C8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—31no—
174C4⋊Q8#6712yes—
175#127—31no—
176#124—61no—
177#121—61no—
178#121—61no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×D4—61no—
184C4×D4—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4#6711yescharacteristic
189C2×C42—21no—
190C2×C42C22×D311yescharacteristic
191#106—24no—
192#106—64no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8#6712yes—
199#103—24no—
200#103—62no—
201#102—62no—
202#102—64no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205C4×C8—61no—
206C4×C8—61no—
207C23×C3#12111yescharacteristic
208C22×D3—41no—
209C3×Q8—22no—
210C3×Q8—22no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—82no—
214C3×Q8—82no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—82no—
223C2×C3×C4—81no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234#67—81no—
235#67—81no—
236C2×Dic3—41no—
237D12—42no—
238D12—41no—
239D12—41no—
240D12—42no—
241C4×D3—81no—
242C4×D3—81no—
243Dic6—41no—
244Dic6—41no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—61no—
261C2×Q8—82no—
262C2×Q8#21512yes—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—121no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—121no—
271C22×C4D3×D411yescharacteristic
272C22×C4#23711yescharacteristic
273C22×C4C2×#6711yescharacteristic
274Q16—124no—
275SD16—124no—
276M16—121no—
277M16—121no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—121no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—41no—
297C42—41no—
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C42C2×#6711yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3#39211yescharacteristic
305C22×C3#46511yescharacteristic
306C22×C3#46811yescharacteristic
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—42no—
314C3×C4—81no—
315C3×C4—82no—
316C3×C4#44811yescharacteristic
317C3×C4#44911yescharacteristic
318C3×C4#45011yescharacteristic
319C3×C4#45211yescharacteristic
320C3×C4#47112yes—
321Dic3—81no—
322C23—121no—
323C23#93911yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—41no—
327Q8—41no—
328Q8—82no—
329Q8—82no—
330Q8—121no—
331Q8—121no—
332D4—122no—
333D4—122no—
334D4—121no—
335D4—121no—
336D4—241no—
337D4—241no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—121no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4#93711yescharacteristic
351C2×C4#93711yescharacteristic
352C2×C4#93811yescharacteristic
353C2×C4#93911yescharacteristic
354C2×C4#94011yescharacteristic
355C2×C4C2×#21511yescharacteristic
356C2×C4#94211yescharacteristic
357C2×C4#94911yescharacteristic
358C2×C4#95011yescharacteristic
359C2×C4#95311yescharacteristic
360C8—42no—
361C8—122no—
362C8—122no—
363C8—241no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#200011yescharacteristic
367C2×C3#200911yescharacteristic
368C2×C3#201911yescharacteristic
369D3—161no—
370C22—21no—
371C22—21no—
372C22—241no—
373C22—241no—
374C22—241no—
375C22#504811yescharacteristic
376C22#504911yescharacteristic
377C22#522311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—81no—
384C4—241no—
385C4#504611yescharacteristic
386C4#504711yescharacteristic
387C4#517712yes—
388C4#522711yescharacteristic
389C4#523811yescharacteristic
390C3#1018911yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—481no—
394C2#7169911yescharacteristic
395C2#7205911yescharacteristic
396C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#475C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#475D3
hypocenterC3#10189
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
4. #71774C2semidirect
5. #71624C2semidirect
7. #67707C2semidirect
33. C2×#475D3semidirect
390. C3#10189semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
#832162
#824412
#717742
#716242
#704262
#677072
#101893