Finite Group Databasebeta

← #10696469

Subgroups — #10696469

Order 768.

658 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C3×#496C211yesmaximal, characteristic
2C2×#5071C211yesmaximal, characteristic, complemented
3#82992C211yesmaximal, characteristic, complemented
4#71765C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic, complemented
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10079—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C22×C3×D8—21no—
11C2×C3×#127C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13C2×#870C2211yescharacteristic
14C3×#496—22no—
15C3×#496—22no—
16C2×C3×#97C2211yescharacteristic
17#5230—21no—
18#5222—21no—
19C2×#832—21no—
20C2×#832—21no—
21C2×#832—22no—
22#5071—22no—
23#5071—22no—
24#4953—21no—
25#4949—21no—
26#4948—22no—
27#4936C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4540—22no—
31#4465—22no—
32C2×#496D311yescharacteristic
33C2×#490—31no—
34#2820—31no—
35#1961—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C2×C3×SD16—24no—
45C2×C3×SD16—44no—
46C2×C3×D8—24no—
47C2×C3×D8—44no—
48C22×C3×C8—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#127D412yes—
53C3×#124—21no—
54C3×#124—21no—
55C3×#124—21no—
56C3×#124—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D412yes—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70#870—21no—
71#870—21no—
72#870—21no—
73#870—21no—
74C3×#97—21no—
75C3×#97D412yes—
76#832—24no—
77#832—24no—
78#832—44no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#803—21no—
83#803—21no—
84#803—21no—
85#803D412yes—
86C22×SD16—21no—
87C22×D8—21no—
88C2×#127C2×D311yescharacteristic
89C2×#125—31no—
90C2×#124C2×D311yescharacteristic
91#496—22no—
92#496—22no—
93#490—62no—
94#490—62no—
95C2×#102—61no—
96C2×#102—61no—
97C2×#102—62no—
98C2×#97—31no—
99C2×#97—31no—
100C2×#97C2×D311yescharacteristic
101#399—61no—
102#397—61no—
103#396—62no—
104#395—61no—
105#394—61no—
106#382—31no—
107#330—62no—
108#329—62no—
109C24×C3—21no—
110C24×C3—41no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—44no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C3×SD16—48no—
136C3×D8—48no—
137C2×C3×C8—24no—
138C2×C3×C8—22no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—82no—
148#217—82no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—42no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—82no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C22×Q8#6711yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4#6711yescharacteristic
169C2×SD16—24no—
170C2×SD16—44no—
171C2×D8—24no—
172C2×D8—44no—
173C22×C8—21no—
174C22×C8—61no—
175C22×C8—61no—
176#127—21no—
177#127—21no—
178#127—21no—
179#127#6712yes—
180#125—61no—
181#125—61no—
182#125—61no—
183#125—61no—
184#124—21no—
185#124—21no—
186#124—21no—
187#124—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—62no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42C22×D311yescharacteristic
199#102—64no—
200#102—64no—
201#102—124no—
202#97—21no—
203#97—32no—
204#97—61no—
205#97—61no—
206#97—61no—
207#97#6712yes—
208#95—62no—
209#95—61no—
210#95—61no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—41no—
217C23×C3—81no—
218C23×C3—82no—
219C23×C3—82no—
220C23×C3—81no—
221C23×C3#12011yescharacteristic
222C3×Q8—42no—
223C3×Q8—81no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—44no—
228C3×D4—44no—
229C3×D4—81no—
230C3×D4—82no—
231C3×D4—82no—
232C3×D4—82no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C3×C8—24no—
254C3⋊C8—42no—
255C3⋊C8—42no—
256C24—21no—
257C24—41no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—21no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4C2×#6711yescharacteristic
282SD16—48no—
283D8—48no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—122no—
289C2×C8—121no—
290C2×C8—121no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—122no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—242no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—242no—
306C22⋊C4—242no—
307C42D3×D411yescharacteristic
308C42#23711yescharacteristic
309C42C2×#6711yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—82no—
314C22×C3—81no—
315C22×C3—82no—
316C22×C3—81no—
317C22×C3—82no—
318C22×C3—81no—
319C22×C3—162no—
320C22×C3—162no—
321C22×C3#39211yescharacteristic
322C22×C3#45311yescharacteristic
323C22×C3#45312yes—
324C22×C3#45412yes—
325C22×C3#45411yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—81no—
333Dic3—82no—
334Dic3—81no—
335Dic3—81no—
336C23—41no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—82no—
344C23—81no—
345C23—82no—
346C23#93711yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—44no—
353D4—44no—
354D4—81no—
355D4—82no—
356D4—82no—
357D4—82no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4#93711yescharacteristic
373C2×C4#93811yescharacteristic
374C2×C4#93911yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4#94011yescharacteristic
377C2×C4#95311yescharacteristic
378C8—24no—
379C8—122no—
380C8—122no—
381C2×C3—161no—
382C2×C3—162no—
383C2×C3#165612yes—
384C2×C3#165712yes—
385C2×C3#200211yescharacteristic
386C2×C3#200311yescharacteristic
387C2×C3#200611yescharacteristic
388C22—82no—
389C22—81no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—82no—
395C22—81no—
396C22—162no—
397C22—162no—
398C22#507712yes—
399C22#519811yescharacteristic
400C22#520112yes—
401C22#520111yescharacteristic
402C22#522311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—241no—
411C4—242no—
412C4—241no—
413C3#1007911yescharacteristic, complemented, Sylow
414C2—161no—
415C2—162no—
416C2#7132712yes—
417C2#7132812yes—
418C2#7176911yescharacteristic
419C2#7209611yescharacteristic
420C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#496C2
socleC23×C3#120
hypercenterC2×#496D3
hypocenterC3#10079
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
2. C2×#5071C2semidirect
3. #82992C2semidirect
4. #71765C2semidirect
5. C2×#4540C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
413. C3#10079semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4962
C2×#50712
#829922
#717652
C2×#45402
C2×#44652
#676592
#100793