Finite Group Databasebeta

← #10697636

Subgroups — #10697636

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
1C3×#3229C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic
3#83239C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#70766C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10251—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#944—21no—
12C2×C4×Dic6C2211yescharacteristic
13C3×#498—21no—
14C3×#487—21no—
15C3×#483—22no—
16C3×#475—22no—
17C3×#431C2211yescharacteristic
18#5229—21no—
19#5094—21no—
20#5091—21no—
21#5063—22no—
22#5004—21no—
23#4995—21no—
24#4983—22no—
25#4972—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4538—22no—
31#4484—22no—
32#3229D311yescharacteristic
33#3078—31no—
34#2546—31no—
35#1546—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×Dic6D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56#944—42no—
57#944—42no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—24no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#106—24no—
70C3×C4⋊C8D412yes—
71C3×#103—24no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—24no—
75C3⋊#95D412yes—
76#830—21no—
77#810—22no—
78#810—22no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#498—21no—
90#487—21no—
91#483—22no—
92#479—62no—
93#477—61no—
94#475—22no—
95#475—61no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423—31no—
100#419—61no—
101#398—62no—
102#391—61no—
103#389—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#328—62no—
107Q8⋊C8—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6SD1612yes—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
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×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C4⋊Dic3—24no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—48no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156#208—41no—
157#208—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×Q8—31no—
162C2×M16—61no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—122no—
168C4⋊Q8—122no—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6712yes—
187C2×C42—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—24no—
192#106—62no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—24no—
198#103—62no—
199#103—62no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C4×C8—21no—
203C4×C8—61no—
204#95—32no—
205#95—64no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×Q8—81no—
212C3×Q8—81no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×C3×C4C2×SD1611yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4#13711yescharacteristic
230C2×Dic3—24no—
231C2×Dic3—22no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238Dic6—22no—
239Dic6—22no—
240Dic6—41no—
241Dic6—41no—
242C3×C8—42no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—32no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—61no—
259C22×C4—22no—
260C22×C4—62no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266M16—121no—
267M16—121no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—64no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—128no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—41no—
288C42—41no—
289C42—121no—
290C42—121no—
291C42—121no—
292C42—121no—
293C42D3×D411yescharacteristic
294C42#23711yescharacteristic
295C42C2×#6711yescharacteristic
296C42C2×#6711yescharacteristic
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—82no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4#44811yescharacteristic
309C3×C4#44911yescharacteristic
310C3×C4#45612yes—
311C3×C4#46511yescharacteristic
312C3×C4#46811yescharacteristic
313Dic3—44no—
314Dic3—42no—
315Dic3—81no—
316C23#93711yescharacteristic
317Q8—42no—
318Q8—62no—
319Q8—62no—
320Q8—81no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—121no—
326Q8—121no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—64no—
335C2×C4—62no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4D3×SD1611yescharacteristic
343C2×C4#91411yescharacteristic
344C2×C4#91511yescharacteristic
345C2×C4#91611yescharacteristic
346C2×C4#93711yescharacteristic
347C2×C4#93811yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#94011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—42no—
353C8—122no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3#200011yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200911yescharacteristic
360C22—21no—
361C22—21no—
362C22#516911yescharacteristic
363C22#517311yescharacteristic
364C22#522311yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—124no—
372C4—122no—
373C4—241no—
374C4#517011yescharacteristic
375C4#517111yescharacteristic
376C4#517211yescharacteristic
377C4#517411yescharacteristic
378C4#524612yes—
379C3#1025111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2#7186811yescharacteristic
383C2#7205411yescharacteristic
384C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3229D3
hypocenterC3#10251
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

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×#32292
#833672
#832392
#718352
#717732
#707662
#677462
#102513