Finite Group Databasebeta

← #10691268

Subgroups — #10691268

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C3⋊C8C211yesmaximal, characteristic
2#82507C211yesmaximal, characteristic
3C3×#1806C211yesmaximal, characteristic
4#70501C211yesmaximal, characteristic, complemented
5#69471C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9207—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×#104—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14#5233—21no—
15Q8×C3⋊C8—21no—
16#5022—21no—
17#5021C411yescharacteristic, complemented
18#5020C411yescharacteristic, complemented
19D4×C3⋊C8—21no—
20D4×C3⋊C8—22no—
21#5009—21no—
22#5008C411yescharacteristic, complemented
23#5007C411yescharacteristic, complemented
24#4998—21no—
25#4925—21no—
26#4925C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4×C3⋊C8—21no—
29C2×C4×C3⋊C8C2211yescharacteristic
30C3×#339—21no—
31#4484—21no—
32#4484—21no—
33#4452—22no—
34#3076—31no—
35C8×#41—31no—
36#1806D311yescharacteristic
37#1503—31no—
38#1493—31no—
39#1301—31no—
40#1279—31no—
41C2×C3×#41D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4×Q8C2×C411yescharacteristic
44C3×C4×Q8D411yescharacteristic
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4D411yescharacteristic
49C3×#117—21no—
50C3×#117D411yescharacteristic
51C2×C3×C4⋊C4—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55#926—21no—
56#924—21no—
57#924—21no—
58C22×C3⋊C8—22no—
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61#870—21no—
62#869C2×C411yescharacteristic
63#869C2×C411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#104—42no—
69C3×#95—21no—
70#835—21no—
71#835—21no—
72#832—21no—
73#832—21no—
74#830—22no—
75#830D412yes—
76#808—21no—
77#808—21no—
78#807—21no—
79#807—21no—
80C4⋊(C3⋊C8)—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)C2×C411yescharacteristic
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4×C3⋊C8—22no—
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8D411yescharacteristic
89C4×C3⋊C8D411yescharacteristic
90#533—31no—
91C4×#41C2×D311yescharacteristic
92#479—61no—
93#478—61no—
94#477—31no—
95#476—31no—
96#475—31no—
97#474—31no—
98C8×Q8—61no—
99C8×D4—62no—
100C8×D4—61no—
101#431—31no—
102#431—61no—
103#427—61no—
104#423C2×D311yescharacteristic
105C2×C4⋊C8—31no—
106C2×#104—21no—
107#416—61no—
108C2×C4×C8—31no—
109C2×C4×C8—61no—
110C4×C4⋊C4C2×D311yescharacteristic
111#339—21no—
112#328—62no—
113Q8⋊C8—61no—
114Q8⋊C8—61no—
115C3×#41—21no—
116C3×#41—21no—
117C3×#41SD1612yes—
118C3×#41Q1612yes—
119C2×C3×Q8C22⋊C411yescharacteristic
120C2×C3×D4—21no—
121C2×C3×D4C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×Dic6—21no—
130C2×Dic6—21no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4C22⋊C411yescharacteristic
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C42—21no—
144C3×C42—21no—
145C3×C42—21no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—44no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×#41#6711yescharacteristic
170C2×M16—21no—
171C22×C8—62no—
172C22×C8—61no—
173C22×C8—61no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176#125—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8C4×D311yescharacteristic
182C4×Q8D1211yescharacteristic
183C4×D4—21no—
184C4×D4—21no—
185C4×D4C4×D311yescharacteristic
186C4×D4D1211yescharacteristic
187#117—21no—
188#117—61no—
189#117#6711yescharacteristic
190C2×C4⋊C4—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194#107—61no—
195#107—61no—
196#106—61no—
197#106—61no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—62no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#104—42no—
206#103—61no—
207#103—61no—
208#102—61no—
209#102—61no—
210C22⋊C8—32no—
211C22⋊C8—62no—
212C4×C8—31no—
213C4×C8—31no—
214C4×C8—62no—
215C4×C8—61no—
216C4×C8—61no—
217#95—21no—
218C23×C3—21no—
219C23×C3#12111yescharacteristic
220C3×Q8—21no—
221C3×Q8#10312yes—
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—21no—
225C3×D4#10312yes—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12311yescharacteristic
246C2×C3×C4C2×SD1611yescharacteristic
247C2×C3×C4C2×Q1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×Dic3—41no—
251C2×Dic3—41no—
252Dic6—41no—
253Dic6—41no—
254Dic6—81no—
255C3×C8—81no—
256C3⋊C8—22no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261#41—21no—
262#41—21no—
263#41#21412yes—
264#41#21612yes—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8(C2×D3)⋊C411yescharacteristic
268C2×D4—21no—
269C2×D4(C2×D3)⋊C411yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277M16—41no—
278M16—41no—
279C2×C8—41no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—62no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—122no—
286C2×C8—124no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4(C2×D3)⋊C411yescharacteristic
300C4⋊C4(C2×D3)⋊C411yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—121no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—41no—
310C42—121no—
311C42C2×C4×D311yescharacteristic
312C42C2×D1211yescharacteristic
313C42D3×D411yescharacteristic
314C42#23711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3#38611yescharacteristic
321C22×C3#46511yescharacteristic
322C22×C3#46711yescharacteristic
323C3×C4—21no—
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—82no—
330C3×C4C2×#10311yescharacteristic
331C3×C4#41611yescharacteristic
332C3×C4#44311yescharacteristic
333C3×C4#44411yescharacteristic
334C3×C4#46511yescharacteristic
335C3×C4#46711yescharacteristic
336Dic3—81no—
337Dic3—81no—
338C23—21no—
339C23#93911yescharacteristic
340Q8—21no—
341Q8—121no—
342Q8—121no—
343Q8—241no—
344Q8#81012yes—
345D4—22no—
346D4—22no—
347D4—21no—
348D4#81012yes—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4#67⋊C411yescharacteristic
366C2×C4#67⋊C411yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88611yescharacteristic
369C2×C4C2×(C2×D3)⋊C411yescharacteristic
370C2×C4C2×#21411yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4C2×#21611yescharacteristic
373C2×C4#95011yescharacteristic
374C2×C4#95011yescharacteristic
375C8—62no—
376C8—81no—
377C8—121no—
378C8—122no—
379C8—121no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3#187611yescharacteristic
385C2×C3#189211yescharacteristic
386C2×C3#190611yescharacteristic
387C22—21no—
388C22—21no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22#495811yescharacteristic
393C22#504111yescharacteristic
394C22#505111yescharacteristic
395C4—22no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—241no—
403C4—241no—
404C4#482311yescharacteristic
405C4#482411yescharacteristic
406C4C2×#81011yescharacteristic
407C4#500211yescharacteristic
408C4#504111yescharacteristic
409C4#505111yescharacteristic
410C3#920711yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—42no—
414C2#6951411yescharacteristic
415C2#7037711yescharacteristic
416C2#7039511yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1806C2
socleC22×C3#386
prime radicalC2×C3×#41D4
cosocleC2×C3×C42C23
hypercenter#1806D3
hypocenterC3#9207
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
4. #70501C2semidirect
5. #69471C2semidirect
6. #67714C2semidirect
7. #67619C2semidirect
17. #5021C4semidirect
18. #5020C4semidirect
22. #5008C4semidirect
23. #5007C4semidirect
410. C3#9207semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊C82
#825072
C3×#18062
#705012
#694712
#677142
#676192
#92073