Finite Group Databasebeta

← #10690909

Subgroups — #10690909

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82810C211yesmaximal, characteristic
2C2×#4648C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9145—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4×Dic3—21no—
11C22×C4×Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5229C412yescomplemented
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4978—22no—
20#4972C412yescomplemented
21#4946—21no—
22#4943—21no—
23#4648—22no—
24#4648—22no—
25C3×#328—22no—
26#4550—22no—
27#4538—22no—
28#2855—31no—
29C2×#376—31no—
30#1817—31no—
31C2×#328—31no—
32C2×#328D311yescharacteristic
33#1294—31no—
34#1294—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C23×Dic3D411yescharacteristic
40C23×Dic3D411yescharacteristic
41C22×C3×C8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×C4⋊Dic3C2×C412yes—
51C2×Dic3⋊C4—22no—
52C2×C4×Dic3—24no—
53C2×C4×Dic3—28no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3C2×C412yes—
59C2×C4×Dic3D412yes—
60C22×C3⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#95—21no—
64C3×#95D412yes—
65C3⋊#95—24no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95D412yes—
72#830—21no—
73#830—21no—
74C2×C22⋊Q8C2×D311yescharacteristic
75C22×C42—31no—
76C22×C42—61no—
77C2×C22⋊C8—31no—
78C2×C22⋊C8C2×D311yescharacteristic
79#391—32no—
80#389—32no—
81#387—62no—
82#385—61no—
83#381—61no—
84#380—31no—
85#376—62no—
86#376—62no—
87C2×#95—31no—
88C2×#95—61no—
89C2×#95C2×D311yescharacteristic
90#336—62no—
91#328—22no—
92#328—62no—
93C24×C3C2×D411yescharacteristic
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—42no—
97C22×C3×C4—24no—
98C22×C3×C4—22no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—24no—
114C22×Dic3—24no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—42no—
124C22×Dic3—41no—
125C22×Dic3—41no—
126C22×Dic3C22⋊C412yes—
127C22×Dic3C22⋊C412yes—
128C2×C3×C8—42no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136#217—41no—
137#217—41no—
138C4⋊Dic3—42no—
139Dic3⋊C4—44no—
140C4×Dic3—216no—
141C4×Dic3—22no—
142C4×Dic3—22no—
143C4×Dic3—22no—
144C4×Dic3—22no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147C22×Q8C4×D311yescharacteristic
148C23×C4—21no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—61no—
152C23×C4C22×D311yescharacteristic
153C22×C8—21no—
154C22×C8—61no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—61no—
165C2×C42—32no—
166C2×C42—32no—
167C2×C42—64no—
168C2×C42—68no—
169C2×C42—61no—
170C2×C42—61no—
171C2×C42—61no—
172C2×C42—61no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176C22⋊C8—61no—
177#95—21no—
178#95—32no—
179#95—62no—
180#95—62no—
181#95—62no—
182#95—64no—
183#95—61no—
184#95—61no—
185#95#6712yes—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3#12011yescharacteristic
190C23×C3#12311yescharacteristic
191C23×C3#13511yescharacteristic
192C23×C3#13511yescharacteristic
193C23×C3#13711yescharacteristic
194C23×C3#13711yescharacteristic
195C3×Q8—81no—
196C3×Q8—81no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—24no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×Dic3—24no—
217C2×Dic3—24no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—24no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—42no—
225C2×Dic3—44no—
226C2×Dic3—44no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3#10414yes—
236C3×C8—81no—
237C3⋊C8—81no—
238C24D3×D411yescharacteristic
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—42no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—64no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—64no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—61no—
263C22×C4—62no—
264C22×C4—122no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4C2×C4×D311yescharacteristic
268C22×C4#23511yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4#23711yescharacteristic
272C22×C4C2×#6711yescharacteristic
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—122no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—122no—
282C4⋊C4—124no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42—616no—
288C42—62no—
289C42—62no—
290C42—62no—
291C42—62no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3#38611yescharacteristic
301C22×C3#41012yes—
302C22×C3#41611yescharacteristic
303C22×C3#41911yescharacteristic
304C22×C3C2×#10412yes—
305C22×C3#45211yescharacteristic
306C22×C3#45211yescharacteristic
307C22×C3#48511yescharacteristic
308C22×C3#48511yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—44no—
314C3×C4—81no—
315C3×C4—81no—
316Dic3—24no—
317Dic3—44no—
318Dic3—42no—
319Dic3—82no—
320C23—22no—
321C23—22no—
322C23D3×C22⋊C411yescharacteristic
323C23#88311yescharacteristic
324C23#91511yescharacteristic
325C23#91611yescharacteristic
326C23#91811yescharacteristic
327C23#91911yescharacteristic
328C23#93711yescharacteristic
329Q8—81no—
330Q8—81no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—34no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—64no—
348C2×C4—64no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—64no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—122no—
356C2×C4—124no—
357C2×C4—124no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4#67⋊C411yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4C4×#6711yescharacteristic
369C2×C4#93811yescharacteristic
370C8—81no—
371C8—241no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3#150012yes—
377C2×C3#150612yes—
378C2×C3#187311yescharacteristic
379C2×C3#189611yescharacteristic
380C2×C3#189911yescharacteristic
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—21no—
388C22—21no—
389C22#474012yes—
390C22#480211yescharacteristic
391C22#480411yescharacteristic
392C22#481411yescharacteristic
393C22#481511yescharacteristic
394C22D3×#10412yes—
395C22#495611yescharacteristic
396C22#517311yescharacteristic
397C22#518711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—64no—
404C4—81no—
405C4—81no—
406C4—124no—
407C4—122no—
408C4—242no—
409C3#914511yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2#6830812yes—
415C2#6831012yes—
416C2#7002911yescharacteristic
417C2#7026911yescharacteristic
418C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9145
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
14. #5229C4semidirect
20. #4972C4semidirect
409. C3#9145semidirect

Maximal subgroups

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

subgroupindex
#828102
C2×#46482
C2×C3×#3282
C2×#45382
#699102
#676752
#676122
#91453