Finite Group Databasebeta

← #10686751

Subgroups — #10686751

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86258C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#70528C211yesmaximal, characteristic
5#70509C211yesmaximal, characteristic
6#69788C211yesmaximal, characteristic
7#69738C211yesmaximal, characteristic
8#8356—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5722—21no—
11#5721—21no—
12#5685—22no—
13#5623C414yes—
14#5622—22no—
15#5567—22no—
16#5567—21no—
17Q8×Dic6—21no—
18#5519—21no—
19#5516—21no—
20#5470C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#423C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×#824—21no—
25#4924C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×Q8⋊C8—22no—
28#4554—21no—
29#4550—22no—
30#4547—21no—
31#4546—21no—
32#4486—21no—
33#4481—21no—
34#4456—22no—
35#3502—31no—
36#1984—31no—
37#1801—31no—
38#1550—31no—
39#1550—31no—
40#1493—31no—
41#1493D311yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8C2×C412yes—
50C3×C4×Q8—21no—
51C3×C4×Q8D412yes—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54Q8×Dic3—22no—
55#944—22no—
56#944—24no—
57#926—21no—
58C2×C4⋊Dic3D411yescharacteristic
59#924—21no—
60#924C2×C412yes—
61C2×Dic3⋊C4C2×C412yes—
62C2×C4×Dic3D412yes—
63C2×#208—21no—
64C22×C3⋊C8—21no—
65#890—22no—
66#889—24no—
67#889—24no—
68#889—24no—
69#888—22no—
70#888—22no—
71#888—24no—
72#887—22no—
73#887—22no—
74#887—24no—
75#887—22no—
76#878—24no—
77#877—24no—
78#870—22no—
79#870—21no—
80#869—22no—
81#869D411yescharacteristic
82#869D411yescharacteristic
83C4×Dic6—22no—
84C4×Dic6—22no—
85C4×Dic6—21no—
86C4×Dic6—21no—
87C4×Dic6—21no—
88C4×Dic6—21no—
89C4×Dic6—21no—
90C4×Dic6D412yes—
91C3×C4⋊C8—21no—
92C3×C4⋊C8—21no—
93C3×C4×C8—21no—
94C3×C4×C8—21no—
95#824—42no—
96C4⋊(C3⋊C8)—21no—
97C4⋊(C3⋊C8)—21no—
98C4⋊(C3⋊C8)—21no—
99C4⋊(C3⋊C8)—21no—
100Q82—61no—
101Q82—61no—
102#557—61no—
103#557—62no—
104#557—61no—
105#557—62no—
106#544—34no—
107#544—61no—
108#533—31no—
109#533—62no—
110#533C2×D311yescharacteristic
111#519—61no—
112C2×C4×Q8—31no—
113#423—31no—
114#423C2×D311yescharacteristic
115C2×C4⋊C8—31no—
116C2×#101—61no—
117C2×C4×C8C2×D311yescharacteristic
118#343—61no—
119#336—62no—
120#333—61no—
121#333—61no—
122#332—61no—
123#332—61no—
124Q8⋊C8—22no—
125Q8⋊C8—62no—
126C2×C3×Q8—21no—
127C2×C3×Q8C22⋊C412yes—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3C22⋊C412yes—
132C22×Dic3C22⋊C412yes—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—24no—
139C2×Dic6C22⋊C412yes—
140C2×Dic6SD1612yes—
141C2×Dic6Q1612yes—
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4C22⋊C412yes—
154C3×C22⋊C4—22no—
155C3×C42—22no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160#217—22no—
161C4⋊Dic3—24no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3C22⋊C412yes—
168Dic3⋊C4—24no—
169Dic3⋊C4—24no—
170Dic3⋊C4—24no—
171Dic3⋊C4—24no—
172Dic3⋊C4—22no—
173Dic3⋊C4—22no—
174Dic3⋊C4—22no—
175Dic3⋊C4—22no—
176C4×Dic3—24no—
177C4×Dic3—24no—
178C4×Dic3—22no—
179C4×Dic3—22no—
180C4×Dic3—22no—
181#208—41no—
182#208—41no—
183C2×C3⋊C8—42no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C22×Q8—31no—
187C2×M16—21no—
188C2×M16—61no—
189C22×C8—21no—
190C22×C8—61no—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193C4⋊Q8—62no—
194C4⋊Q8—64no—
195C4⋊Q8—62no—
196C4⋊Q8—62no—
197C4⋊Q8—62no—
198C4⋊Q8—64no—
199C4⋊Q8D1211yescharacteristic
200C4⋊Q8D1211yescharacteristic
201#125—21no—
202#125—62no—
203#125—64no—
204#125—64no—
205#125—62no—
206#125—64no—
207#125—61no—
208C22⋊Q8—21no—
209C22⋊Q8—32no—
210C22⋊Q8—64no—
211C22⋊Q8—61no—
212C22⋊Q8C4×D312yes—
213C4×Q8—21no—
214C4×Q8—32no—
215C4×Q8—62no—
216C4×Q8—62no—
217C4×Q8—62no—
218C4×Q8—62no—
219C4×Q8—64no—
220C4×Q8—62no—
221C4×Q8—62no—
222C4×Q8—61no—
223C4×Q8—61no—
224C4×Q8—61no—
225C4×Q8—61no—
226C4×Q8—61no—
227C4×Q8#6712yes—
228#117—21no—
229#117—64no—
230#117—61no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—31no—
233C2×C42—32no—
234C2×C42C22×D311yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—61no—
238C4⋊C8—61no—
239C4⋊C8—61no—
240C4⋊C8—61no—
241#101—122no—
242C4×C8—21no—
243C4×C8—21no—
244C23×C3C2×C22⋊C411yescharacteristic
245C3×Q8—42no—
246C3×Q8—41no—
247C3×Q8—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4C2×C22⋊C411yescharacteristic
254C2×C3×C4C2×C22⋊C411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4C2×SD1611yescharacteristic
260C2×C3×C4C2×Q1611yescharacteristic
261C2×C3×C4#13711yescharacteristic
262C2×C3×C4#13711yescharacteristic
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—42no—
269C2×Dic3#10314yes—
270Dic6—22no—
271Dic6—22no—
272Dic6—22no—
273Dic6—22no—
274Dic6—41no—
275Dic6—41no—
276Dic6—42no—
277Dic6—84no—
278C3×C8—42no—
279C3×C8—81no—
280C3⋊C8—81no—
281C3⋊C8—81no—
282C2×Q8—21no—
283C2×Q8—32no—
284C2×Q8—32no—
285C2×Q8—32no—
286C2×Q8—64no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×Q8—61no—
290C2×Q8—61no—
291C2×Q8—61no—
292C2×Q8(C2×D3)⋊C412yes—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4C2×C4×D311yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298M16—41no—
299M16—41no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—122no—
307C2×C8—121no—
308C2×C8—121no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—32no—
315C4⋊C4—64no—
316C4⋊C4—64no—
317C4⋊C4—64no—
318C4⋊C4—64no—
319C4⋊C4—64no—
320C4⋊C4—62no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—62no—
329C4⋊C4(C2×D3)⋊C412yes—
330C22⋊C4—22no—
331C22⋊C4—62no—
332C42—22no—
333C42—64no—
334C42—64no—
335C42—62no—
336C42—62no—
337C42—62no—
338C42C2×D1211yescharacteristic
339C42D3×D411yescharacteristic
340C42D3×D411yescharacteristic
341C42C2×#6711yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3#37911yescharacteristic
345C22×C3C2×#10311yescharacteristic
346C22×C3#41911yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—42no—
351C3×C4—42no—
352C3×C4#44811yescharacteristic
353C3×C4#44811yescharacteristic
354C3×C4#45111yescharacteristic
355C3×C4#45111yescharacteristic
356C3×C4#45411yescharacteristic
357C3×C4#45611yescharacteristic
358Dic3—24no—
359Dic3—42no—
360Dic3—42no—
361Dic3—42no—
362C23D3×C22⋊C411yescharacteristic
363Q8—42no—
364Q8—41no—
365Q8—41no—
366Q8—62no—
367Q8—62no—
368Q8—62no—
369Q8—62no—
370Q8—122no—
371Q8—121no—
372Q8—121no—
373Q8—244no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—34no—
379C2×C4—41no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—122no—
386C2×C4D3×C22⋊C411yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#88211yescharacteristic
389C2×C4D3×SD1611yescharacteristic
390C2×C4#91511yescharacteristic
391C2×C4D3×Q1611yescharacteristic
392C2×C4#91811yescharacteristic
393C2×C4C2×(C2×D3)⋊C411yescharacteristic
394C2×C4#93711yescharacteristic
395C2×C4#93711yescharacteristic
396C8—42no—
397C8—81no—
398C8—241no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3#178711yescharacteristic
403C2×C3#179111yescharacteristic
404C2×C3#179311yescharacteristic
405C22—21no—
406C22—21no—
407C22D3×#10311yescharacteristic
408C22#480211yescharacteristic
409C22#495511yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—64no—
416C4—122no—
417C4—122no—
418C4—122no—
419C4#480611yescharacteristic
420C4#480911yescharacteristic
421C4#482611yescharacteristic
422C4#482711yescharacteristic
423C4#517011yescharacteristic
424C4#518611yescharacteristic
425C3#835611yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2#7000911yescharacteristic
429C2#7026611yescharacteristic
430C2#7038711yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#379
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#8356
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
425. C3#8356semidirect

Maximal subgroups

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

subgroupindex
#862582
C3×#14932
#718302
#705282
#705092
#697882
#697382
#83563