Finite Group Databasebeta

← #10691022

Subgroups — #10691022

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82315C211yesmaximal, characteristic
2#82303C211yesmaximal, characteristic
3C3×#1762C211yesmaximal, characteristic
4#69801C211yesmaximal, characteristic, complemented
5#69471C211yesmaximal, characteristic, complemented
6#67662C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9165—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×#102—21no—
12C2×C3×#97C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5218—21no—
15#5207—21no—
16C2×#830—21no—
17#5019C411yescharacteristic, complemented
18#5019C411yescharacteristic, complemented
19#5018—21no—
20#5018—21no—
21#5012—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#5007C411yescharacteristic, complemented
25#5007C411yescharacteristic, complemented
26#4970—21no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C3×#340—21no—
31#4465—22no—
32#4452—21no—
33#4452—21no—
34#3110—31no—
35#2974—31no—
36#1762D311yescharacteristic
37#1531—31no—
38#1503—31no—
39#1302—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×C3×C8—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4C2×C411yescharacteristic
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4D411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#214—21no—
56C2×#214—21no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—22no—
61C22×C3⋊C8—22no—
62#870—21no—
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#102—24no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#95—21no—
71#832—21no—
72#832—21no—
73#830—22no—
74#830—22no—
75#830—21no—
76#830—21no—
77#830D411yescharacteristic
78#830D411yescharacteristic
79#810—21no—
80#810—21no—
81#807—21no—
82#807—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#533—31no—
91C2×C4×D4C2×D311yescharacteristic
92#484—61no—
93#474—31no—
94#474—31no—
95#461—31no—
96#461—31no—
97#450—61no—
98#440—61no—
99#440—61no—
100#436—61no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432—31no—
105#431—31no—
106C2×#102—21no—
107#407—61no—
108C2×C22⋊C8—61no—
109C2×#97C2×D311yescharacteristic
110C4×C4⋊C4C2×D311yescharacteristic
111#340—21no—
112#330—62no—
113#328—61no—
114#328—61no—
115C24×C3—21no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4C22⋊C411yescharacteristic
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—82no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C42—81no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—41no—
151#214—42no—
152#214—42no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160#208—82no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—82no—
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C2×SD16—61no—
170C2×SD16—61no—
171C2×M16—62no—
172C22×C8—21no—
173C22×C8—62no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176#125—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4C4×D311yescharacteristic
184C4×D4C4×D311yescharacteristic
185C4×D4D1211yescharacteristic
186C4×D4D1211yescharacteristic
187#117—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C42—21no—
192C2×C42C22×D311yescharacteristic
193#107—61no—
194#107—61no—
195C4⋊C8—31no—
196C4⋊C8—31no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199#103—61no—
200#103—61no—
201#102—24no—
202#102—61no—
203#102—61no—
204C22⋊C8—31no—
205C22⋊C8—31no—
206C22⋊C8—62no—
207C22⋊C8—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#97—21no—
211#97—21no—
212#97—32no—
213#97—61no—
214C4×C8—61no—
215#95—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—81no—
222C23×C3#12111yescharacteristic
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—21no—
226C3×D4—42no—
227C3×D4#10412yes—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×Dic3—41no—
250C2×Dic3—41no—
251Dic6—41no—
252Dic6—41no—
253Dic6—81no—
254C3×C8—42no—
255C3⋊C8—41no—
256C3⋊C8—42no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C24—21no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4(C2×D3)⋊C411yescharacteristic
267C2×D4(C2×D3)⋊C411yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278SD16—122no—
279SD16—122no—
280M16—242no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—242no—
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—21no—
305C22⋊C4—121no—
306C42—81no—
307C42—121no—
308C42C2×C4×D311yescharacteristic
309C42C2×D1211yescharacteristic
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3#38611yescharacteristic
323C22×C3#47111yescharacteristic
324C22×C3#47311yescharacteristic
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—82no—
333C3×C4C2×#10411yescharacteristic
334C3×C4#42111yescharacteristic
335Dic3—81no—
336Dic3—81no—
337C23—21no—
338C23—21no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23#93911yescharacteristic
344Q8—121no—
345Q8—121no—
346Q8—241no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4#80512yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4#67⋊C411yescharacteristic
370C2×C4#67⋊C411yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4#88611yescharacteristic
373C2×C4C2×(C2×D3)⋊C411yescharacteristic
374C2×C4#93911yescharacteristic
375C8—42no—
376C8—121no—
377C8—122no—
378C8—121no—
379C8—121no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3—81no—
385C2×C3#187111yescharacteristic
386C2×C3#190411yescharacteristic
387C2×C3#190811yescharacteristic
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—82no—
397C22—82no—
398C22#495811yescharacteristic
399C22#520011yescharacteristic
400C22#520411yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—82no—
409C4—241no—
410C4—241no—
411C4C2×#80511yescharacteristic
412C4#500511yescharacteristic
413C3#916511yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#6945311yescharacteristic
419C2#7030411yescharacteristic
420C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1762C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1762D3
hypocenterC3#9165
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
4. #69801C2semidirect
5. #69471C2semidirect
6. #67662C2semidirect
7. #67621C2semidirect
17. #5019C4semidirect
18. #5019C4semidirect
24. #5007C4semidirect
25. #5007C4semidirect
413. C3#9165semidirect

Maximal subgroups

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

subgroupindex
#823152
#823032
C3×#17622
#698012
#694712
#676622
#676212
#91653