Finite Group Databasebeta

← #10690824

Subgroups — #10690824

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83269C211yesmaximal, characteristic
2C2×#4833C211yesmaximal, characteristic
3C2×C3×#333C211yesmaximal, characteristic, complemented
4C2×#4546C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic
6#67702C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9128—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946C412yes—
11C2×C42×D3C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5229C412yes—
15C2×(C2×D3)⋊C8—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×C8×Dic3—21no—
18#4987—21no—
19#4987—21no—
20#4972—21no—
21#4972—21no—
22#4935C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4833—22no—
26#4833—22no—
27C3×#333—22no—
28#4550—21no—
29#4546—22no—
30C2×#431—31no—
31#2857—31no—
32#1797—31no—
33C2×#333—31no—
34C2×#333D311yescharacteristic, complemented
35#1296—31no—
36#1296—31no—
37C22×C3×Q8C2×C412yes—
38C22×C4×D3C2×C411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#946—44no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2×C412yes—
55C2×Dic3⋊C4—22no—
56C2×C4×Dic3C2×C411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61C42×D3—21no—
62C42×D3—21no—
63C42×D3—21no—
64C42×D3—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71(C2×D3)⋊C8—24no—
72(C2×D3)⋊C8—42no—
73C8×Dic3—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C22⋊Q8—32no—
80C22×C42—31no—
81#431—62no—
82#431—62no—
83C2×C4⋊C8—31no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C22⋊C8—61no—
88C2×C22⋊C8—61no—
89C2×C4×C8—61no—
90#391—32no—
91#389—61no—
92#389—61no—
93#384—31no—
94#380—61no—
95#380—61no—
96#336—61no—
97#333—22no—
98#333—62no—
99C23×D3C22⋊C411yescharacteristic
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—22no—
125C2×C4×D3—41no—
126C2×C4×D3SD1612yes—
127C2×C4×D3Q1612yes—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—44no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—42no—
148Dic3⋊C4—44no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8C4×D312yes—
162C23×C4—31no—
163C23×C4—31no—
164C23×C4—31no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—61no—
168C22×C8—61no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C22⋊Q8—124no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—62no—
181C2×C42—31no—
182C2×C42—31no—
183C2×C42—31no—
184C2×C42—61no—
185C2×C42—61no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197C22⋊C8—64no—
198C22⋊C8—122no—
199C4×C8—64no—
200#95—62no—
201#95—62no—
202C23×C3C2×C22⋊C411yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3#10312yes—
210C3×Q8—82no—
211C3×Q8—82no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×C3×C4C2×SD1611yescharacteristic
224C2×C3×C4C2×Q1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4#13711yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3#10012yes—
242C2×Dic3#10312yes—
243C4×D3—22no—
244C4×D3—22no—
245C4×D3—22no—
246C4×D3—22no—
247C4×D3—42no—
248C4×D3—42no—
249C4×D3—41no—
250C4×D3—41no—
251C4×D3—41no—
252C4×D3—41no—
253C3×C8—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—31no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—42no—
262C22×C4—22no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—31no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—62no—
293C2×C8—121no—
294C2×C8—121no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—121no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—122no—
303C4⋊C4—124no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—124no—
309C42—61no—
310C42—61no—
311C42—61no—
312C42—61no—
313C42—61no—
314C42—61no—
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3#38611yescharacteristic
320C22×C3C2×#10011yescharacteristic
321C22×C3#41311yescharacteristic
322C22×C3C2×#10311yescharacteristic
323C22×C3#41611yescharacteristic
324C22×C3#41611yescharacteristic
325C22×C3#41911yescharacteristic
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—82no—
336C3×C4#45211yescharacteristic
337C3×C4#45211yescharacteristic
338C3×C4#46511yescharacteristic
339C3×C4#46511yescharacteristic
340C3×C4#46711yescharacteristic
341C3×C4#46711yescharacteristic
342C3×C4#48511yescharacteristic
343C3×C4#48511yescharacteristic
344Dic3—22no—
345Dic3—22no—
346Dic3—41no—
347Dic3—82no—
348C23—32no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23D3×C22⋊C411yescharacteristic
356Q8—82no—
357Q8—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—32no—
361C2×C4—32no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4#67⋊C411yescharacteristic
386C2×C4#88311yescharacteristic
387C2×C4#88411yescharacteristic
388C2×C4D3×SD1611yescharacteristic
389C2×C4#91511yescharacteristic
390C2×C4#91611yescharacteristic
391C2×C4#91611yescharacteristic
392C2×C4D3×Q1611yescharacteristic
393C2×C4#91811yescharacteristic
394C2×C4#91911yescharacteristic
395C2×C4#91911yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#93811yescharacteristic
399C8—42no—
400C8—81no—
401C8—122no—
402C8—241no—
403C2×C3#154211yescharacteristic
404C2×C3#154311yescharacteristic
405C2×C3#154811yescharacteristic
406C2×C3#155411yescharacteristic
407C2×C3#187311yescharacteristic
408C2×C3#189211yescharacteristic
409C2×C3#190011yescharacteristic
410D3—42no—
411C22—62no—
412C22—62no—
413C22—62no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22D3×#10011yescharacteristic
419C22#474511yescharacteristic
420C22D3×#10311yescharacteristic
421C22#480211yescharacteristic
422C22#480411yescharacteristic
423C22#480411yescharacteristic
424C22#495611yescharacteristic
425C4—21no—
426C4—21no—
427C4—62no—
428C4—62no—
429C4—82no—
430C4—121no—
431C4—242no—
432C4#480711yescharacteristic
433C4#481311yescharacteristic
434C4#481411yescharacteristic
435C4#481511yescharacteristic
436C4#517111yescharacteristic
437C4#517311yescharacteristic
438C4#518711yescharacteristic
439C4#518811yescharacteristic
440C3#912811yescharacteristic, complemented, Sylow
441C2—122no—
442C2#6848111yescharacteristic
443C2#6848211yescharacteristic
444C2#6848311yescharacteristic
445C2#6848611yescharacteristic
446C2#7008111yescharacteristic
447C2#7026911yescharacteristic
448C2#7027011yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#333C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC2×#333D3
hypocenterC3#9128
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. C2×C3×#333C2semidirect
4. C2×#4546C2semidirect
6. #67702C2semidirect
7. #67702C2semidirect
34. C2×#333D3semidirect
440. C3#9128semidirect

Maximal subgroups

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

subgroupindex
#832692
C2×#48332
C2×C3×#3332
C2×#45462
#704752
#677022
#91283