Finite Group Databasebeta

← #10696021

Subgroups — #10696021

Order 768.

496 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 83 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#83369C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic
3#73727C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic
5#69863C211yesmaximal, characteristic
6#69857C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#10011—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5094—21no—
15#5090—21no—
16#5082—21no—
17#5081—21no—
18#5050—21no—
19#5044—21no—
20#5043—21no—
21#5043—21no—
22#5030—22no—
23#5022—22no—
24#4984—21no—
25#4984—21no—
26#4928—22no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4661—21no—
30#4656—21no—
31C3×#339—22no—
32#4465—22no—
33#4456—22no—
34#3236—31no—
35#3113—31no—
36#2584—31no—
37#1986D311yescharacteristic
38#1528—31no—
39#1491—31no—
40#1285—31no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
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×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C42D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×#216—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×#95—21no—
75#835—22no—
76#835—22no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×Q8—31no—
93#501—61no—
94#491—61no—
95#484—61no—
96#483—61no—
97#483—61no—
98#479—62no—
99#475—61no—
100#462—61no—
101#451—61no—
102#441—62no—
103#432—31no—
104#432—31no—
105C4×M16C2×D311yescharacteristic
106#398—61no—
107#395—61no—
108#389—61no—
109#389—61no—
110#389—62no—
111C4×C4⋊C4C2×D311yescharacteristic
112#339—22no—
113#330—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C3×M16—22no—
129C3×M16—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—24no—
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×C22⋊C4—41no—
139C3×C42—42no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#216—44no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—44no—
153C4⋊Dic3—82no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C22×Q8—31no—
165C2×Q16—62no—
166C2×M16—21no—
167C4⋊Q8D1211yescharacteristic
168C4⋊Q8D1211yescharacteristic
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—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—62no—
192C2×C42—61no—
193C2×C42#6712yes—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#106—62no—
197#106—62no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200#103—62no—
201#103—62no—
202#103—62no—
203#103—62no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—21no—
207#97—32no—
208#97—61no—
209C4×C8—21no—
210#95—21no—
211#95—62no—
212#95—62no—
213C23×C3#12011yescharacteristic
214C3×Q8—42no—
215C3×Q8—42no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242Dic6—22no—
243Dic6—22no—
244Dic6—41no—
245Dic6—41no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—122no—
254C22×C4—22no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—124no—
263M16—22no—
264M16—41no—
265C2×C8—41no—
266C2×C8—62no—
267C2×C8—62no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—124no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—242no—
289C22⋊C4—41no—
290C42—42no—
291C42—41no—
292C42—121no—
293C42—121no—
294C42C2×D1211yescharacteristic
295C42#23511yescharacteristic
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
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—44no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#45612yes—
312Dic3—42no—
313Dic3—81no—
314Dic3—81no—
315Dic3—81no—
316C23#93711yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—62no—
320Q8—62no—
321Q8—121no—
322Q8—121no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—62no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4#88211yescharacteristic
344C2×C4#88211yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4#88411yescharacteristic
347C2×C4#93011yescharacteristic
348C2×C4#94011yescharacteristic
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#496011yescharacteristic
360C22#506111yescharacteristic
361C22#509511yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—81no—
369C4—81no—
370C4—122no—
371C4—241no—
372C4—241no—
373C4—241no—
374C4#482612yes—
375C3#1001111yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2#6969411yescharacteristic
379C2#7038911yescharacteristic
380C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1986D3
hypocenterC3#10011
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

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
#833692
#833262
#737272
C3×#19862
#698632
#698572
#676212
#100113