Finite Group Databasebeta

← #10697222

Subgroups — #10697222

Order 768.

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

normal subgroups only

This group has 496 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
1C3×#3078C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic
3C2×#5031C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic
8#10187—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C22×#216—21no—
12C2×#869C2211yescharacteristic
13C3×#479—22no—
14C3×#477—21no—
15C3×#475—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#419—21no—
19#5080—21no—
20#5079—21no—
21#5043—22no—
22#5031—22no—
23#5031—22no—
24#5020—22no—
25C2×#810—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—22no—
30#4528—22no—
31#4482—22no—
32#3229—31no—
33#3078D311yescharacteristic
34C2×#462—31no—
35#1546—31no—
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59C2×#216—42no—
60C2×#216—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C22×C3⋊C8—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
68C3×#107—22no—
69C3×#106—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#103—22no—
73C3×#103—22no—
74#835—24no—
75#830—21no—
76#830D412yes—
77#810—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—
85C22×Q16—61no—
86#533C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#479—22no—
93#477—21no—
94#475—21no—
95#475—62no—
96#462—62no—
97#462—62no—
98#431—31no—
99#431—31no—
100#428—21no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#419—21no—
104C2×#103—61no—
105#332—62no—
106Q8⋊C8—62no—
107#324—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8SD1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—41no—
122C2×Dic6—22no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C3×M16—41no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149#216—48no—
150C4⋊Dic3—22no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—61no—
160C22×Q8#6711yescharacteristic
161C2×Q16—64no—
162C2×Q16—122no—
163C2×Q16—122no—
164C2×M16—21no—
165C22×C8—61no—
166C4⋊Q8—32no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
172#125—21no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181#117—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#107—22no—
189#106—22no—
190#106—64no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—22no—
198#103—22no—
199#103—64no—
200#103—64no—
201C22⋊C8—32no—
202C22⋊C8—61no—
203C4×C8—61no—
204C4×C8—61no—
205C23×C3#12111yescharacteristic
206C3×Q8—22no—
207C3×Q8—22no—
208C3×Q8—42no—
209C3×Q8—41no—
210C3×Q8—41no—
211C3×Q8—42no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—42no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234Dic6—42no—
235Dic6—42no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—62no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8#21512yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—121no—
259C22×C4D3×D411yescharacteristic
260C22×C4#23711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263M16—41no—
264M16—41no—
265C2×C8—22no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—121no—
269C2×C8—121no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C42C2×#6711yescharacteristic
292C42C2×#6711yescharacteristic
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—42no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4#44811yescharacteristic
306C3×C4#44911yescharacteristic
307C3×C4#45011yescharacteristic
308C3×C4#45211yescharacteristic
309C3×C4#47312yes—
310Dic3—82no—
311C23#93911yescharacteristic
312Q8—22no—
313Q8—22no—
314Q8—42no—
315Q8—41no—
316Q8—41no—
317Q8—42no—
318Q8—122no—
319Q8—122no—
320Q8—241no—
321Q8—241no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93811yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#94011yescharacteristic
339C2×C4C2×#21511yescharacteristic
340C2×C4#94211yescharacteristic
341C2×C4#94911yescharacteristic
342C2×C4#95011yescharacteristic
343C2×C4#95311yescharacteristic
344C8—42no—
345C8—122no—
346C8—122no—
347C8—241no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#200011yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#202011yescharacteristic
353C22—21no—
354C22—21no—
355C22#504811yescharacteristic
356C22#504911yescharacteristic
357C22#522311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—242no—
366C4#504611yescharacteristic
367C4#504711yescharacteristic
368C4#520412yes—
369C4#522711yescharacteristic
370C4#523811yescharacteristic
371C3#1018711yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2#7169911yescharacteristic
375C2#7210911yescharacteristic
376C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3078D3
hypocenterC3#10187
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

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
C3×#30782
#833232
C2×#50312
#718242
#716242
#705062
#677142
#101873