Finite Group Databasebeta

← #10696482

Subgroups — #10696482

Order 768.

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

normal subgroups only

This group has 648 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×#489C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic, complemented
3#83007C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10081—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C22×#216—21no—
12C22×#215—21no—
13C2×#875C2211yescharacteristic
14C3×#489—22no—
15C3×#489—22no—
16C2×C3×#103—21no—
17C2×C3×#102—21no—
18C2×C3×#97C2211yescharacteristic
19#5231—21no—
20#5223—21no—
21#5086—22no—
22#5086—22no—
23#4961—22no—
24#4960—21no—
25#4947—21no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4540—22no—
30#4529—22no—
31#4464—22no—
32C2×#497—31no—
33C2×#489D311yescharacteristic, complemented
34#2819—31no—
35#1993—31no—
36C2×#329—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×#67—22no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yescomplemented
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#216—24no—
61C2×#216—44no—
62C2×#215—24no—
63C2×#215—44no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3D412yes—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70#875—21no—
71#875—21no—
72#875—21no—
73#875—21no—
74C3×#103—24no—
75C3×#102—24no—
76C3×#97—21no—
77C3×#97D412yescomplemented
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#803—21no—
82#803—21no—
83#803—21no—
84#803D412yescomplemented
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89C2×#124C2×D311yescharacteristic
90#497—62no—
91#497—62no—
92#489—22no—
93#489—22no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#394—62no—
100#393—61no—
101#393—61no—
102#392—61no—
103#392—61no—
104#382—31no—
105#340—62no—
106#329—62no—
107#329—62no—
108C24×C3—21no—
109C23×D3—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×Q8—41no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×#67—82no—
126C2×#67—82no—
127C2×#67—82no—
128C22×Dic3—22no—
129C22×Dic3—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—42no—
136C2×C3×C8—22no—
137C2×C3×C8—42no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—82no—
151#216—48no—
152#215—48no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—82no—
158C4⋊Dic3—42no—
159C2×C3⋊C8—24no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C22×Q8—21no—
165C22×Q8—31no—
166C22×Q8#6711yescharacteristic
167C22×D4—31no—
168C22×D4—62no—
169C22×D4#6711yescharacteristic
170C2×Q16—64no—
171C2×Q16—124no—
172C2×SD16—64no—
173C2×SD16—124no—
174C22×C8—21no—
175C22×C8—61no—
176C22×C8—61no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8#6712yescomplemented
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—61no—
186#124—61no—
187#124—61no—
188#124—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—32no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42C22×D311yescharacteristic
199#103—24no—
200#102—24no—
201#97—21no—
202#97—32no—
203#97—61no—
204#97—61no—
205#97—61no—
206#97#6712yescomplemented
207#95—62no—
208#95—61no—
209#95—61no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C22×D3—41no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—42no—
224C3×Q8—81no—
225C3×Q8—82no—
226C3×D4—42no—
227C3×D4—81no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242#67—88no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248D12—42no—
249D12—81no—
250Dic6—42no—
251Dic6—81no—
252C3×C8—42no—
253C3⋊C8—24no—
254C3⋊C8—42no—
255C24—21no—
256C24—61no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—122no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—42no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—122no—
273C2×D4—242no—
274C2×D4—242no—
275C2×D4—242no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4C2×#6711yescharacteristic
283Q16—128no—
284SD16—128no—
285C2×C8—22no—
286C2×C8—42no—
287C2×C8—64no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—122no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—242no—
306C22⋊C4—242no—
307C42D3×D411yescharacteristic
308C42#23711yescharacteristic
309C42C2×#6711yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#39211yescharacteristic
317C22×C3#45312yes—
318C22×C3#45511yescharacteristic
319C22×C3#45512yes—
320C22×C3#45611yescharacteristic
321C2×D3—82no—
322C2×D3—82no—
323C2×D3—81no—
324C2×D3—81no—
325C2×D3—81no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—81no—
334Dic3—82no—
335Dic3—81no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—122no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—122no—
346C23#93711yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—82no—
352Q8—122no—
353Q8—241no—
354D4—42no—
355D4—81no—
356D4—122no—
357D4—241no—
358D4—248no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4#93711yescharacteristic
373C2×C4#93811yescharacteristic
374C2×C4#93911yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4#94011yescharacteristic
377C2×C4#95311yescharacteristic
378C8—42no—
379C8—64no—
380C8—122no—
381C2×C3—161no—
382C2×C3#165912yes—
383C2×C3#166112yes—
384C2×C3#200211yescharacteristic
385C2×C3#200411yescharacteristic
386C2×C3#200711yescharacteristic
387D3—161no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—81no—
392C22—81no—
393C22—241no—
394C22—242no—
395C22—242no—
396C22—241no—
397C22—241no—
398C22#506112yes—
399C22#519812yes—
400C22#520211yescharacteristic
401C22#520511yescharacteristic
402C22#522311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4—242no—
412C4—241no—
413C3#1008111yescharacteristic, complemented, Sylow
414C2—161no—
415C2—481no—
416C2#7124412yes—
417C2#7124512yes—
418C2#7173111yescharacteristic
419C2#7209411yescharacteristic
420C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#489C2
socleC23×C3#120
prime radical#83007C2
hypercenterC2×#489D3
hypocenterC3#10081
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×C3×#489C2semidirect
2. C2×#5086C2semidirect
4. #71788C2semidirect
5. C2×#4540C2semidirect
6. C2×#4464C2semidirect
7. #67659C2semidirect
33. C2×#489D3semidirect
49. C3×C4⋊Q8D4semidirect
77. C3×#97D4semidirect
84. #803D4semidirect
180. C4⋊Q8#67semidirect
206. #97#67semidirect
413. C3#10081semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
C2×#50862
#830072
#717882
C2×#45402
C2×#44642
#676592
#100813