Finite Group Databasebeta

← #10697225

Subgroups — #10697225

Order 768.

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

normal subgroups only

This group has 496 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
1#79271C211yesmaximal, characteristic
2#79068C211yesmaximal, characteristic
3C2×#4849C211yesmaximal, characteristic
4#70786C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#70429C211yesmaximal, characteristic
7C3×#1301C211yesmaximal, characteristic
8#10187—31nomaximal, Sylow
9C22×Dic12—21no—
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16#5121—21no—
17#5120—21no—
18C2×#816—21no—
19#4925C2211yescharacteristic
20#4851—22no—
21#4851—21no—
22#4850—21no—
23#4849—22no—
24#4849—22no—
25#4829—22no—
26#4721—22no—
27#4681—21no—
28#4680—21no—
29#4484—22no—
30#4482—22no—
31#4469—22no—
32#3229—31no—
33#3078—31no—
34C2×#462—31no—
35#1546—31no—
36#1481—31no—
37#1461—31no—
38#1301D311yescharacteristic
39C22×Dic6—21no—
40C22×Dic6D411yescharacteristic
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×Dic12—24no—
53C2×Dic12—42no—
54C2×Dic12—42no—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D412yes—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8D412yes—
74C3×C4×C8—21no—
75#830—21no—
76#818—22no—
77#817—22no—
78#817—24no—
79#816—22no—
80#816—22no—
81#816—24no—
82#816—24no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85C22×Q16—61no—
86#533—31no—
87C2×C4⋊Q8—31no—
88C2×C4×Q8—31no—
89#498—61no—
90#487—61no—
91#483—62no—
92#479—62no—
93#477—61no—
94#475—61no—
95#475—62no—
96#462—62no—
97#462—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8C2×D311yescharacteristic
103#419—61no—
104C2×#103—61no—
105#332—62no—
106Q8⋊C8—62no—
107#324—62no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—41no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6SD1612yes—
128C3×M16—41no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139#217—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158Dic12—48no—
159C22×Q8—31no—
160C22×Q8—61no—
161C2×Q16—64no—
162C2×Q16—122no—
163C2×Q16—122no—
164C2×M16—21no—
165C22×C8—21no—
166C4⋊Q8—32no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172#125—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181#117—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#106—62no—
190#106—64no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8D1212yes—
196C4⋊C8D1212yes—
197#103—64no—
198#103—62no—
199#103—62no—
200#103—64no—
201C22⋊C8—61no—
202C22⋊C8D1212yes—
203C4×C8—21no—
204C4×C8—61no—
205C23×C3#12111yescharacteristic
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×C3×C4C2×SD1611yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4#13711yescharacteristic
218C2×Dic3—22no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228Dic6—22no—
229Dic6—22no—
230Dic6—41no—
231Dic6—41no—
232Dic6—42no—
233Dic6—42no—
234Dic6—42no—
235Dic6—42no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—42no—
239C3×C8—42no—
240C3×C8—42no—
241C3⋊C8—81no—
242C2×Q8—32no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—62no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—121no—
259C22×C4C2×D1211yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4#23911yescharacteristic
262Q16—128no—
263M16—41no—
264M16—41no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—121no—
271C4⋊C4—61no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C42—121no—
287C42—121no—
288C42—121no—
289C42C2×D1211yescharacteristic
290C42C2×D1211yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#39211yescharacteristic
296C22×C3#46511yescharacteristic
297C22×C3#46811yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4#44811yescharacteristic
302C3×C4#44911yescharacteristic
303C3×C4#45011yescharacteristic
304C3×C4#45211yescharacteristic
305C3×C4#47312yes—
306Dic3—42no—
307Dic3—82no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23#89511yescharacteristic
312Q8—62no—
313Q8—62no—
314Q8—122no—
315Q8—121no—
316Q8—121no—
317Q8—122no—
318Q8—122no—
319Q8—122no—
320Q8—241no—
321Q8—241no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#87411yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88211yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#89511yescharacteristic
340C2×C4#91111yescharacteristic
341C2×C4D3×SD1611yescharacteristic
342C2×C4#91611yescharacteristic
343C2×C4#91811yescharacteristic
344C8—42no—
345C8—42no—
346C8—42no—
347C8—241no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#200011yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#202011yescharacteristic
353C22—21no—
354C22—21no—
355C22#467111yescharacteristic
356C22#486411yescharacteristic
357C22#488611yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—122no—
362C4—242no—
363C4—241no—
364C4—241no—
365C4—241no—
366C4#477611yescharacteristic
367C4#478111yescharacteristic
368C4#480611yescharacteristic
369C4#480811yescharacteristic
370C4#489612yes—
371C3#1018711yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2#6888711yescharacteristic
375C2#6894411yescharacteristic
376C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1301D3
hypocenterC3#10187
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
371. C3#10187semidirect

Maximal subgroups

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

subgroupindex
#792712
#790682
C2×#48492
#707862
#705062
#704292
C3×#13012
#101873