Finite Group Databasebeta

← #10697641

Subgroups — #10697641

Order 768.

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

normal subgroups only

This group has 524 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
3#73711C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic
8#10251—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869—21no—
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5121—21no—
17#5120—21no—
18#4984—22no—
19#4929—21no—
20#4925C2211yescharacteristic
21#4851—21no—
22#4851—22no—
23#4850—21no—
24#4829—22no—
25#4721—22no—
26#4681—21no—
27#4680—21no—
28#4661—22no—
29#4484—22no—
30#4482—22no—
31#4452—22no—
32#3229—31no—
33#3078—31no—
34#2546—31no—
35#1546—31no—
36#1503—31no—
37#1479—31no—
38#1303D311yescharacteristic
39C22×Dic6—21no—
40C22×Dic6D411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#926—21no—
46C2×C4⋊Dic3—24no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D412yes—
50#924—21no—
51#924—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—42no—
60#869—42no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3×#95D412yes—
75C3⋊#95—24no—
76#830—21no—
77#818—22no—
78#817—22no—
79#817—24no—
80#816—22no—
81#816—22no—
82#816—24no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×C4⋊Q8—31no—
87C2×C4⋊Q8—61no—
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#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423C2×D311yescharacteristic
100#419—61no—
101#398—62no—
102#391—61no—
103#389—62no—
104C4×C4⋊C4C2×D311yescharacteristic
105#332—62no—
106#328—62no—
107Q8⋊C8—62no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6SD1612yes—
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C3×C4⋊C4—24no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—48no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—41no—
160C22×Q8—31no—
161C22×Q8—61no—
162C2×M16—21no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—122no—
170C4⋊Q8—122no—
171#125—61no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192#106—64no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8D1212yes—
196C4⋊C8D1212yes—
197#103—62no—
198#103—62no—
199#103—64no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C4×C8—21no—
203C4×C8—61no—
204#95—64no—
205#95D1212yes—
206C23×C3#12011yescharacteristic
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×C3×C4C2×SD1611yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232Dic6—22no—
233Dic6—22no—
234Dic6—41no—
235Dic6—41no—
236Dic6—42no—
237Dic6—81no—
238Dic6—81no—
239Dic6—81no—
240Dic6—81no—
241Dic6—81no—
242C3×C8—42no—
243C3×C8—42no—
244C3⋊C8—81no—
245C2×Q8—32no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8—121no—
259C22×C4—22no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—62no—
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266M16—41no—
267M16—41no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—121no—
271C4⋊C4—24no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—62no—
279C4⋊C4—61no—
280C4⋊C4—128no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—121no—
288C42—41no—
289C42—41no—
290C42—121no—
291C42—121no—
292C42—121no—
293C42C2×D1211yescharacteristic
294C42C2×D1211yescharacteristic
295C42D3×D411yescharacteristic
296C42#23911yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#39211yescharacteristic
300C22×C3#45011yescharacteristic
301C22×C3#45211yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—44no—
306C3×C4#44811yescharacteristic
307C3×C4#44911yescharacteristic
308C3×C4#45612yes—
309C3×C4#46511yescharacteristic
310C3×C4#46811yescharacteristic
311Dic3—42no—
312Dic3—82no—
313Dic3—81no—
314Dic3—81no—
315Dic3—81no—
316C23#88211yescharacteristic
317Q8—62no—
318Q8—62no—
319Q8—121no—
320Q8—121no—
321Q8—122no—
322Q8—241no—
323Q8—241no—
324Q8—241no—
325Q8—241no—
326Q8—241no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—62no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4#87411yescharacteristic
343C2×C4#88211yescharacteristic
344C2×C4#88211yescharacteristic
345C2×C4#89511yescharacteristic
346C2×C4#89511yescharacteristic
347C2×C4#89511yescharacteristic
348C2×C4#91111yescharacteristic
349C2×C4D3×SD1611yescharacteristic
350C2×C4#91611yescharacteristic
351C2×C4#91811yescharacteristic
352C8—42no—
353C8—42no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3#200011yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200911yescharacteristic
360C22—21no—
361C22—21no—
362C22#467111yescharacteristic
363C22#477611yescharacteristic
364C22#480811yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—122no—
370C4—242no—
371C4—241no—
372C4—241no—
373C4—241no—
374C4#478111yescharacteristic
375C4#480611yescharacteristic
376C4#482612yes—
377C4#486411yescharacteristic
378C4#488611yescharacteristic
379C3#1025111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2#6876811yescharacteristic
383C2#6888711yescharacteristic
384C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1303D3
hypocenterC3#10251
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
379. C3#10251semidirect

Maximal subgroups

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

subgroupindex
#792712
#790682
#737112
#707662
#705062
#694712
C3×#13032
#102513