Finite Group Databasebeta

← #10697212

Subgroups — #10697212

Order 768.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3229C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3C2×#5083C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70783C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic, complemented
8#10186—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#498—21no—
14C3×#487—21no—
15C3×#483—22no—
16C3×#475—22no—
17C3×#431C2211yescharacteristic
18C2×#835—21no—
19#5087—21no—
20#5084—21no—
21#5083—22no—
22#5083—22no—
23#5066—22no—
24#5001—21no—
25#4995—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4547—22no—
30#4535—22no—
31#4468—22no—
32#3229D311yescharacteristic, complemented
33#3077—31no—
34C2×#461—31no—
35#1546—31no—
36#1481—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#215—24no—
58C2×#215—42no—
59C2×#215—42no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×#106—24no—
69C3×C4⋊C8—21no—
70C3×#103—24no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#835—24no—
74#830—21no—
75#809—22no—
76#809—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C22×SD16—61no—
86#533C2×D311yescharacteristic
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×D4—31no—
89#498—21no—
90#487—21no—
91#483—22no—
92#478—62no—
93#476—61no—
94#475—22no—
95#474—61no—
96#461—62no—
97#461—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423—31no—
102C2×C4⋊C8—31no—
103#418—61no—
104C2×#103—61no—
105#332—62no—
106Q8⋊C8—62no—
107#323—62no—
108C23×D3—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12SD1612yes—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—82no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#215—48no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×D4—31no—
162C23×C4—61no—
163C2×SD16—64no—
164C2×SD16—122no—
165C2×SD16—122no—
166C2×M16—61no—
167C22×C8—61no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8#6712yes—
172C4⋊Q8#6711yescharacteristic
173C4⋊Q8#6711yescharacteristic
174#125—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183#117—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C22⋊C4—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—24no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198#103—24no—
199#103—64no—
200#102—62no—
201#102—62no—
202C22⋊C8—61no—
203C22⋊C8#6712yes—
204C4×C8—21no—
205C4×C8—61no—
206C23×C3#12111yescharacteristic
207C22×D3—22no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—81no—
211C22×D3—81no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—42no—
216C3×Q8—81no—
217C3×Q8—81no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—41no—
236C2×Dic3—81no—
237D12—22no—
238D12—22no—
239D12—41no—
240D12—41no—
241C4×D3—82no—
242C4×D3—82no—
243C3×C8—42no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C24—61no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C22×C4—41no—
263C22×C4—61no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—242no—
267C22×C4D3×D411yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270SD16—128no—
271M16—121no—
272M16—121no—
273C2×C8—22no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—121no—
277C2×C8—121no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C22⋊C4—41no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—41no—
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42C2×#6711yescharacteristic
298C42C2×#6711yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3#39211yescharacteristic
302C22×C3#46511yescharacteristic
303C22×C3#46811yescharacteristic
304C2×D3—42no—
305C2×D3—82no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—82no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4#44811yescharacteristic
316C3×C4#44911yescharacteristic
317C3×C4#45011yescharacteristic
318C3×C4#45211yescharacteristic
319C3×C4#47112yes—
320Dic3—81no—
321C23—62no—
322C23—121no—
323C23—121no—
324C23—241no—
325C23—241no—
326C23#93811yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—81no—
332Q8—81no—
333D4—62no—
334D4—62no—
335D4—121no—
336D4—121no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—121no—
345C2×C4—242no—
346C2×C4—242no—
347C2×C4—241no—
348C2×C4D3×SD1611yescharacteristic
349C2×C4#91411yescharacteristic
350C2×C4#91511yescharacteristic
351C2×C4#91611yescharacteristic
352C2×C4#93711yescharacteristic
353C2×C4#93711yescharacteristic
354C2×C4#93911yescharacteristic
355C2×C4#93911yescharacteristic
356C2×C4#94011yescharacteristic
357C2×C4#95311yescharacteristic
358C8—42no—
359C8—122no—
360C8—122no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#200011yescharacteristic
365C2×C3#200911yescharacteristic
366C2×C3#201911yescharacteristic
367D3—82no—
368C22—21no—
369C22—21no—
370C22—122no—
371C22—242no—
372C22—242no—
373C22—241no—
374C22—241no—
375C22#517111yescharacteristic
376C22#517411yescharacteristic
377C22#522311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—241no—
385C4#516911yescharacteristic
386C4#517011yescharacteristic
387C4#517211yescharacteristic
388C4#517311yescharacteristic
389C4#520312yes—
390C3#1018611yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#7186911yescharacteristic
395C2#7205411yescharacteristic
396C2#7210711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3229D3
hypocenterC3#10186
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C3×#3229C2semidirect
4. #71824C2semidirect
5. #71773C2semidirect
7. #67745C2semidirect
32. #3229D3semidirect
390. C3#10186semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833502
C2×#50832
#718242
#717732
#707832
#677452
#101863