Finite Group Databasebeta

← #10695966

Subgroups — #10695966

Order 768.

500 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2581C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic
6#69546C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#10003—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#400—21no—
13C3×#395—21no—
14C3×#391—21no—
15C3×#389—21no—
16C3×#387—22no—
17C3×#382C2211yescharacteristic
18#5228—21no—
19#5225—21no—
20#5031—21no—
21C4×#216—22no—
22#5022—22no—
23#5021—21no—
24#5021—21no—
25#5020—21no—
26#4970—21no—
27#4964—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32#4535—22no—
33#4452—22no—
34#3104—31no—
35#3078—31no—
36#2581D311yescharacteristic
37#1518—31no—
38#1487—31no—
39#1479—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#216—22no—
61#926—21no—
62#924—21no—
63#924—21no—
64C2×#208—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73C3×#95—21no—
74C3×#95D412yes—
75#835—22no—
76#835—22no—
77#835—22no—
78#830—21no—
79#830—21no—
80#810—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93#484—61no—
94#479—62no—
95#477—61no—
96#477—61no—
97#475—61no—
98#462—61no—
99#451—61no—
100C4×Q16—62no—
101#431—31no—
102#431—31no—
103#428—61no—
104#423—31no—
105#419—61no—
106#400—21no—
107#395—21no—
108#391—21no—
109#389—21no—
110#387—22no—
111#382C2×D311yescharacteristic
112#328—62no—
113Q8⋊C8—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×Q8Q1612yescomplemented
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×Dic6—21no—
133C2×Dic6—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—44no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—41no—
154#216—44no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—41no—
162#208—41no—
163#208—41no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8#6711yescharacteristic
168C22×Q8#6711yescharacteristic
169C2×Q16—62no—
170C2×M16—61no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173#125—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—61no—
185C4×Q8—61no—
186#117—61no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C4⋊C4#6711yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#107—62no—
198#106—62no—
199C4⋊C8—32no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203#103—62no—
204#103—62no—
205#103—62no—
206#103—62no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C4×C8—32no—
210C4×C8—61no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95#6712yes—
216C23×C3#12011yescharacteristic
217C3×Q8—22no—
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—22no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×Q8—41no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×Q1611yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4#13711yescharacteristic
249C2×C3×C4#13711yescharacteristic
250C2×Dic3—41no—
251C2×Dic3—41no—
252Dic6—42no—
253Dic6—81no—
254C3⋊C8—42no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8#21612yescomplemented
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4C2×#6711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C22×C4C2×#6711yescharacteristic
276Q16—124no—
277M16—121no—
278M16—121no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—44no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—121no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—121no—
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42#23711yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3#39211yescharacteristic
312C22×C3#45111yescharacteristic
313C22×C3#45211yescharacteristic
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4#45111yescharacteristic
324C3×C4#45211yescharacteristic
325C3×C4#46711yescharacteristic
326C3×C4#47011yescharacteristic
327Dic3—81no—
328Dic3—81no—
329C23#95311yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—122no—
339Q8—241no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4#93711yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#93811yescharacteristic
360C2×C4#93911yescharacteristic
361C2×C4#93911yescharacteristic
362C2×C4#94011yescharacteristic
363C2×C4#94211yescharacteristic
364C2×C4C2×#21611yescharacteristic
365C2×C4#95011yescharacteristic
366C2×C4#95111yescharacteristic
367C8—122no—
368C8—122no—
369C8—241no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#200111yescharacteristic
373C2×C3#200911yescharacteristic
374C2×C3#201011yescharacteristic
375C22—21no—
376C22—21no—
377C22#522311yescharacteristic
378C22#522811yescharacteristic
379C22#523911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—241no—
390C4—241no—
391C4#504711yescharacteristic
392C4#505011yescharacteristic
393C4#505111yescharacteristic
394C4#505211yescharacteristic
395C3#1000311yescharacteristic, complemented, Sylow
396C2—21no—
397C2—21no—
398C2#7170011yescharacteristic
399C2#7171511yescharacteristic
400C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2581C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2581D3
hypocenterC3#10003
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
123. C2×C3×Q8Q16semidirect
267. C2×Q8#216semidirect
395. C3#10003semidirect

Maximal subgroups

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

subgroupindex
C3×#25812
#824472
#824312
#716222
#716212
#695462
#676312
#100033