Finite Group Databasebeta

← #10697638

Subgroups — #10697638

Order 768.

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

normal subgroups only

This group has 524 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×#2546C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10251—31nomaximal, Sylow
9C2×C3×C4⋊Q8—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#398—22no—
14C3×#391—21no—
15C3×#389—22no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5225—21no—
18#5094—22no—
19#5093—21no—
20#5080—21no—
21#5022—22no—
22#5021—21no—
23#5020—21no—
24#4971—22no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4535—22no—
31#4452—22no—
32#3229—31no—
33#3078—31no—
34#2546D311yescharacteristic
35#1546—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—42no—
46C3×C4⋊Q8—42no—
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—24no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D412yes—
57C2×C3×C42—21no—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60#926—21no—
61#924—21no—
62#924—21no—
63C2×#208—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#95—24no—
70C3×#95D412yes—
71#835—22no—
72#835—22no—
73#830—21no—
74#830—21no—
75#810—24no—
76#808—22no—
77#808—24no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#479—62no—
93#477—61no—
94#475—62no—
95#475—61no—
96#431—31no—
97#431—31no—
98#428—61no—
99#423—31no—
100#419—61no—
101#398—22no—
102#391—21no—
103#389—22no—
104C4×C4⋊C4C2×D311yescharacteristic
105#332—62no—
106#328—62no—
107Q8⋊C8—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8SD1612yes—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—48no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×Q8#6711yescharacteristic
162C2×M16—61no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171#125—61no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—61no—
179C4×Q8—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C4⋊C4#6712yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192#106—64no—
193C4⋊C8—32no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—64no—
198#103—62no—
199#103—62no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C4×C8—61no—
203C4×C8—61no—
204#95—24no—
205#95#6712yes—
206C23×C3#12011yescharacteristic
207C3×Q8—22no—
208C3×Q8—22no—
209C3×Q8—41no—
210C3×Q8—41no—
211C3×Q8—81no—
212C3×Q8—81no—
213C3×Q8—81no—
214C3×Q8—81no—
215C2×C3×C4—22no—
216C2×C3×C4—24no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234C2×C3×C4C2×SD1611yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×Dic3—41no—
239C2×Dic3—41no—
240Dic6—42no—
241Dic6—81no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8#21412yes—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4C2×#6711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266M16—121no—
267M16—121no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—48no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—121no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—121no—
293C42D3×D411yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C42#23711yescharacteristic
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—42no—
307C3×C4—82no—
308C3×C4—81no—
309C3×C4#44811yescharacteristic
310C3×C4#44911yescharacteristic
311C3×C4#45612yes—
312C3×C4#46511yescharacteristic
313C3×C4#46811yescharacteristic
314Dic3—81no—
315Dic3—81no—
316C23#95311yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—41no—
320Q8—41no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—122no—
326Q8—241no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4#93211yescharacteristic
343C2×C4C2×#21411yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#93711yescharacteristic
346C2×C4#93811yescharacteristic
347C2×C4#93911yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4#94011yescharacteristic
350C2×C4#95011yescharacteristic
351C2×C4#95111yescharacteristic
352C8—122no—
353C8—122no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3#200011yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200911yescharacteristic
360C22—21no—
361C22—21no—
362C22#521811yescharacteristic
363C22#522311yescharacteristic
364C22#523911yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—42no—
370C4—82no—
371C4—81no—
372C4—241no—
373C4—241no—
374C4#503711yescharacteristic
375C4#504011yescharacteristic
376C4#504111yescharacteristic
377C4#504211yescharacteristic
378C4#509512yes—
379C3#1025111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2#7168511yescharacteristic
383C2#7179511yescharacteristic
384C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2546C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2546D3
hypocenterC3#10251
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

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
C3×#25462
#832232
#824312
#717732
#716212
#694712
#676202
#102513