Finite Group Databasebeta

← #10696019

Subgroups — #10696019

Order 768.

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3113C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic
3#82483C211yesmaximal, characteristic
4#71841C211yesmaximal, characteristic
5#71623C211yesmaximal, characteristic
6#70812C211yesmaximal, characteristic
7#67749C211yesmaximal, characteristic
8#10011—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#483—21no—
13C3×#475—21no—
14C3×#462—21no—
15C3×#451—21no—
16C3×#441—22no—
17C3×#432C2211yescharacteristic
18#5230—21no—
19#5082—21no—
20#5081—21no—
21#5044—21no—
22#5043—21no—
23#5022—22no—
24#4983—21no—
25#4976—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4928—21no—
28#4928—21no—
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32#4532—22no—
33#4465—22no—
34#3236—31no—
35#3113D311yescharacteristic
36#2584—31no—
37#1986—31no—
38#1528—31no—
39#1491—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C2×C3×Q16—22no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—22no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#870—21no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#103—22no—
76C3×C22⋊C8—21no—
77C3×#97D412yes—
78#835—22no—
79#835—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#830—21no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×Q8C2×D311yescharacteristic
93#501—61no—
94#491—61no—
95#484—61no—
96#483—21no—
97#483—61no—
98#479—62no—
99#475—21no—
100#462—21no—
101#451—21no—
102#441—22no—
103#432—31no—
104#432C2×D311yescharacteristic
105C4×M16—31no—
106#398—61no—
107#395—61no—
108#389—61no—
109#389—61no—
110#389—62no—
111C4×C4⋊C4—31no—
112#339—62no—
113#330—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C3×Q16—44no—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148C4⋊Dic3—24no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—44no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—82no—
159C4×Dic3—42no—
160#208—22no—
161#208—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C22×Q8#6711yescharacteristic
165C2×Q16—22no—
166C2×M16—61no—
167C4⋊Q8#6711yescharacteristic
168C4⋊Q8#6711yescharacteristic
169#125—21no—
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42—32no—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#106—22no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—32no—
200#103—22no—
201#103—22no—
202#103—62no—
203#103—62no—
204C22⋊C8—21no—
205C22⋊C8—61no—
206#97—61no—
207#97—61no—
208#97#6712yes—
209C4×C8—61no—
210#95—62no—
211#95—62no—
212#95—61no—
213C23×C3#12011yescharacteristic
214C3×Q8—22no—
215C3×Q8—22no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×Q8—42no—
219C3×Q8—42no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238C2×Dic3—24no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—42no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—44no—
263M16—62no—
264M16—121no—
265C2×C8—22no—
266C2×C8—62no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—64no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—124no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—242no—
289C22⋊C4—41no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—122no—
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C2×#6711yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#39211yescharacteristic
301C22×C3#45511yescharacteristic
302C22×C3#45611yescharacteristic
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—42no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4#45612yes—
313Dic3—44no—
314Dic3—81no—
315Dic3—81no—
316C23#93711yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—41no—
320Q8—41no—
321Q8—42no—
322Q8—42no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—64no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4#93711yescharacteristic
344C2×C4#93811yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#94011yescharacteristic
348C2×C4#95311yescharacteristic
349C8—42no—
350C8—122no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#200511yescharacteristic
355C2×C3#200711yescharacteristic
356C2×C3#201011yescharacteristic
357C22—21no—
358C22—21no—
359C22#520211yescharacteristic
360C22#520511yescharacteristic
361C22#522311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—124no—
372C4—241no—
373C4—241no—
374C4#520512yes—
375C3#1001111yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2#7171511yescharacteristic
379C2#7210811yescharacteristic
380C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3113C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3113D3
hypocenterC3#10011
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
375. C3#10011semidirect

Maximal subgroups

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

subgroupindex
C3×#31132
#833262
#824832
#718412
#716232
#708122
#677492
#100113