Finite Group Databasebeta

← #10695968

Subgroups — #10695968

Order 768.

500 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 99 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#82481C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3#79310C211yesmaximal, characteristic
4C3×#1479C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic
6#70770C211yesmaximal, characteristic
7#67738C211yesmaximal, characteristic
8#10003—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5229—21no—
15#5225—21no—
16#5022—22no—
17#5021—21no—
18#5020—21no—
19#4986—21no—
20#4979—21no—
21#4972—21no—
22#4964—21no—
23#4936C2211yescharacteristic
24#4933—22no—
25#4924C2211yescharacteristic
26#4850—21no—
27#4849—21no—
28#4739—21no—
29#4722—21no—
30C4×Dic12—22no—
31C3×Q8⋊C8—22no—
32#4467—22no—
33#4456—22no—
34#3104—31no—
35#3078—31no—
36#2581—31no—
37#1518—31no—
38#1487—31no—
39#1479D311yescharacteristic
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51#926—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54#924—21no—
55#924—21no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×#208—21no—
60C2×Dic12—22no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75C3×C4×C8D412yes—
76#835—22no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#818—22no—
84#816—22no—
85#816—22no—
86#808—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90#533—31no—
91C2×C4×Q8—31no—
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#431C2×D311yescharacteristic
102#431C2×D311yescharacteristic
103#428—61no—
104#423—31no—
105#419—61no—
106#400—61no—
107#395—61no—
108#391—61no—
109#389—61no—
110#387—62no—
111#382—31no—
112#328—62no—
113Q8⋊C8—22no—
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×Q8Q1612yescomplemented
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—42no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#217—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155Dic3⋊C4—44no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166Dic12—44no—
167C22×Q8—31no—
168C22×Q8D1211yescharacteristic
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—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186#117—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4D1211yescharacteristic
193C2×C42—21no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#107—62no—
198#106—62no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—32no—
202C4⋊C8—61no—
203#103—62no—
204#103—62no—
205#103—62no—
206#103—62no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C4×C8—21no—
210C4×C8#6712yes—
211#95—32no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—61no—
216C23×C3#12011yescharacteristic
217C3×Q8—22no—
218C3×Q8—22no—
219C3×Q8—41no—
220C3×Q8—41no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
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×Q1611yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×Dic3—22no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248Dic6—22no—
249Dic6—22no—
250Dic6—41no—
251Dic6—41no—
252Dic6—42no—
253Dic6—81no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—122no—
267C2×Q8Dic1212yescomplemented
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—61no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276Q16—124no—
277M16—121no—
278M16—121no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—62no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—124no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—121no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42—121no—
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42C2×#6711yescharacteristic
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—81no—
320C3×C4#45111yescharacteristic
321C3×C4#45211yescharacteristic
322C3×C4#46711yescharacteristic
323C3×C4#47011yescharacteristic
324Dic3—42no—
325Dic3—82no—
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329C23#88211yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—41no—
333Q8—41no—
334Q8—62no—
335Q8—62no—
336Q8—121no—
337Q8—121no—
338Q8—122no—
339Q8—241no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—62no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4#88211yescharacteristic
358C2×C4#88411yescharacteristic
359C2×C4#88411yescharacteristic
360C2×C4#90411yescharacteristic
361C2×C4C2×Dic1211yescharacteristic
362C2×C4#90911yescharacteristic
363C2×C4#90911yescharacteristic
364C2×C4#93011yescharacteristic
365C2×C4#93711yescharacteristic
366C2×C4#94011yescharacteristic
367C8—42no—
368C8—81no—
369C8—122no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#200111yescharacteristic
373C2×C3#200911yescharacteristic
374C2×C3#201011yescharacteristic
375C22—21no—
376C22—21no—
377C22#473311yescharacteristic
378C22#473911yescharacteristic
379C22#496011yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—122no—
387C4—242no—
388C4—241no—
389C4—241no—
390C4—241no—
391C4#473311yescharacteristic
392C4#473911yescharacteristic
393C4#511611yescharacteristic
394C4#513711yescharacteristic
395C3#1000311yescharacteristic, complemented, Sylow
396C2—21no—
397C2—21no—
398C2#6996411yescharacteristic
399C2#7035111yescharacteristic
400C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1479D3
hypocenterC3#10003
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
120. C2×C3×Q8Q16semidirect
267. C2×Q8Dic12semidirect
395. C3#10003semidirect

Maximal subgroups

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

subgroupindex
#824812
#824312
#793102
C3×#14792
#707932
#707702
#677382
#100033