Finite Group Databasebeta

← #10690949

Subgroups — #10690949

Order 768.

586 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82810C211yesmaximal, characteristic
2C2×#4651C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9153—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4×Dic3C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5229C412yescomplemented
14#5212C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4978—22no—
20#4972C412yescomplemented
21#4946—21no—
22#4943—21no—
23#4651—22no—
24#4651—22no—
25C3×#328—22no—
26#4550—22no—
27#4538—22no—
28#2855—31no—
29C2×#383—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—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3C2×C412yes—
57C2×C4×Dic3D412yes—
58C22×C3⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95D412yes—
63C3⋊#95—24no—
64C3⋊#95—24no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95D412yes—
71#830—21no—
72#830—21no—
73C2×C22⋊Q8C2×D311yescharacteristic
74C22×C42—31no—
75C2×C22⋊C8—31no—
76C2×C22⋊C8C2×D311yescharacteristic
77#391—32no—
78#389—32no—
79#387—62no—
80#385—61no—
81#383—62no—
82#383—62no—
83#381—61no—
84#380—31no—
85C2×#95—31no—
86C2×#95—61no—
87C2×#95—61no—
88C2×#95C2×D311yescharacteristic
89#336—62no—
90#328—22no—
91#328—62no—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—42no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3C22⋊C412yes—
126C22×Dic3C22⋊C412yes—
127C2×C3×C8—42no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135#217—41no—
136#217—41no—
137C4⋊Dic3—42no—
138Dic3⋊C4—44no—
139C4×Dic3—22no—
140C4×Dic3—22no—
141C4×Dic3—22no—
142C4×Dic3—22no—
143C2×C3⋊C8—42no—
144C2×C3⋊C8—41no—
145C22×Q8C4×D311yescharacteristic
146C23×C4—21no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—61no—
150C23×C4C22×D311yescharacteristic
151C22×C8—21no—
152C22×C8—61no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—61no—
163C2×C42—32no—
164C2×C42—32no—
165C2×C42—61no—
166C2×C42—61no—
167C2×C42—61no—
168C2×C42—61no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#95—21no—
174#95—32no—
175#95—62no—
176#95—62no—
177#95—62no—
178#95—64no—
179#95—64no—
180#95—61no—
181#95—61no—
182#95#6712yes—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3C2×C22⋊C411yescharacteristic
186C23×C3#12011yescharacteristic
187C23×C3#12311yescharacteristic
188C23×C3#13511yescharacteristic
189C23×C3#13511yescharacteristic
190C23×C3#13711yescharacteristic
191C23×C3#13711yescharacteristic
192C3×Q8—81no—
193C3×Q8—81no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4—82no—
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—82no—
232C2×Dic3—82no—
233C3×C8—81no—
234C3⋊C8—81no—
235C24D3×D411yescharacteristic
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—42no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—32no—
243C22×C4—32no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—122no—
260C22×C4—122no—
261C22×C4—122no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4C2×C4×D311yescharacteristic
265C22×C4#23511yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—122no—
273C2×C8—121no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—122no—
279C4⋊C4—124no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C42—62no—
285C42—62no—
286C42—62no—
287C42—62no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3#38611yescharacteristic
297C22×C3#41012yes—
298C22×C3#41611yescharacteristic
299C22×C3#41911yescharacteristic
300C22×C3#42112yes—
301C22×C3#45211yescharacteristic
302C22×C3#45211yescharacteristic
303C22×C3#48511yescharacteristic
304C22×C3#48511yescharacteristic
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4—81no—
311C3×C4—81no—
312Dic3—42no—
313Dic3—42no—
314Dic3—82no—
315Dic3—82no—
316C23—22no—
317C23—22no—
318C23D3×C22⋊C411yescharacteristic
319C23#88311yescharacteristic
320C23#91511yescharacteristic
321C23#91611yescharacteristic
322C23#91811yescharacteristic
323C23#91911yescharacteristic
324C23#93711yescharacteristic
325Q8—81no—
326Q8—81no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—242no—
361C2×C4—242no—
362C2×C4#67⋊C411yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4C4×#6711yescharacteristic
365C2×C4#93811yescharacteristic
366C8—81no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#151512yes—
373C2×C3#152112yes—
374C2×C3#187311yescharacteristic
375C2×C3#189711yescharacteristic
376C2×C3#189911yescharacteristic
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22#474012yes—
386C22#480211yescharacteristic
387C22#480411yescharacteristic
388C22#481411yescharacteristic
389C22#481511yescharacteristic
390C22#482112yes—
391C22#495611yescharacteristic
392C22#517311yescharacteristic
393C22#518711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—122no—
402C4—122no—
403C4—242no—
404C4—242no—
405C3#915311yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#6831312yes—
411C2#6831512yes—
412C2#7002911yescharacteristic
413C2#7026911yescharacteristic
414C2#7038311yescharacteristic
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#9153
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
13. #5229C4semidirect
20. #4972C4semidirect
405. C3#9153semidirect

Maximal subgroups

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

subgroupindex
#828102
C2×#46512
C2×C3×#3282
C2×#45382
#699102
#676752
#676122
#91533