Finite Group Databasebeta

← #10777937

Subgroups — #10777937

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81792C211yesmaximal, characteristic, complemented
2#80949C212yesmaximal
3#79517C211yesmaximal, characteristic
4C3×#1689C211yesmaximal, characteristic, complemented
5#71309C211yesmaximal, characteristic, complemented
6#71106C212yesmaximal, complemented
7#70873C211yesmaximal, characteristic, complemented
8#68552C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68476C212yesmaximal, complemented
11#68419C212yesmaximal, complemented
12#19161—31nomaximal, Sylow
13#5728C2212yes—
14C4×#237C2211yescharacteristic
15C3×#494C2211yescharacteristic
16C3×#474C2212yes—
17C3×#436C2211yescharacteristic
18#5188—21no—
19#5188—21no—
20Dic3×Q16—21no—
21Dic3×Q16—21no—
22#5127—21no—
23Dic3×M16—21no—
24#5094C2212yes—
25#5079C2212yes—
26#5019C2212yes—
27#5012C2211yescharacteristic
28#4888—22no—
29#4873—22no—
30#4849C2212yes—
31#4833C2212yes—
32#4833C2211yescharacteristic
33#4831C2211yescharacteristic
34#4783—22no—
35#4758—22no—
36#4728—21no—
37#4718—21no—
38C4×D3×C8C2211yescharacteristic
39#4682C2211yescharacteristic
40#4679C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#332C2212yes—
43#4547C2212yes—
44#4486C2212yes—
45#4482C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3170—31no—
49#3107—32no—
50#2965—31no—
51#1700—32no—
52#1699—32no—
53#1694—31no—
54#1693—31no—
55#1690—31no—
56#1689D311yescharacteristic, complemented
57#1564—31no—
58#1544—32no—
59C2×#237D411yescharacteristic
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C2×C3×M16—21no—
63C3×C4⋊Q8C2312yes—
64C3×C4×D4C2311yescharacteristic
65#946—22no—
66Q8×Dic3—22no—
67D4×Dic3D411yescharacteristic
68C2×#214—22no—
69C4×#67—21no—
70C2×C4×Dic3—21no—
71C2×#208—21no—
72C2×Dic12—22no—
73C2×D3×C8D411yescharacteristic
74C2×D3×C8D411yescharacteristic
75#897—22no—
76#892—22no—
77#892D411yescharacteristic
78#890—22no—
79#887—22no—
80#887D411yescharacteristic
81#67⋊C4—21no—
82#877—21no—
83C42×D3C2311yescharacteristic
84#869C2312yes—
85C4×Dic6C2311yescharacteristic
86C3×#107—22no—
87C3×C4⋊C8C2312yes—
88C3×C4⋊C8C2311yescharacteristic
89C3×#102—22no—
90C3×C22⋊C8—21no—
91C3×C4×C8C2311yescharacteristic
92#835—22no—
93#835—22no—
94#830—21no—
95(C2×D3)⋊C8—21no—
96(C2×D3)⋊C8—22no—
97#818—21no—
98#818—21no—
99#816—22no—
100#815—21no—
101#815—21no—
102Dic3⋊C8—21no—
103Dic3⋊C8—21no—
104C8×Dic3—22no—
105C8×Dic3—21no—
106C8×Dic3D411yescharacteristic
107C8×Dic3D411yescharacteristic
108#810—22no—
109#810—22no—
110#808—22no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8C2311yescharacteristic
114#533—32no—
115C4×#41—31no—
116#500—31no—
117#494C2×D311yescharacteristic
118#487—32no—
119#485—62no—
120#475—32no—
121#474C2×D312yes—
122#467—61no—
123#467—61no—
124#462—32no—
125#461—32no—
126#452—62no—
127#446—31no—
128#446—31no—
129C4×Q16—61no—
130C4×Q16—61no—
131C4×Q16—62no—
132C4×SD16—62no—
133#436—31no—
134#436C2×D311yescharacteristic
135#435—61no—
136#435—61no—
137#431—31no—
138#431—32no—
139#431—61no—
140C4×M16—61no—
141C2×C4×C8—31no—
142#335—32no—
143#334—31no—
144#334C2×D311yescharacteristic
145#332—32no—
146#332—32no—
147#332—32no—
148#332C2×D312yes—
149C2×C3×Q8—22no—
150C2×C3×D4C2×D411yescharacteristic
151C22×C3×C4—21no—
152C2×#67—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155#237—42no—
156C2×C4×D3C2×D411yescharacteristic
157C2×C4×D3C2×D411yescharacteristic
158C2×C4×D3#4111yescharacteristic
159C2×Dic6—22no—
160C2×Dic6C2×D411yescharacteristic
161C3×Q16—42no—
162C3×Q16—42no—
163C3×M16—42no—
164C2×C3×C8—21no—
165C2×C3×C8—22no—
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2×D411yescharacteristic
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D411yescharacteristic
171C3×C22⋊C4—21no—
172C3×C42C2411yescharacteristic
173#217—21no—
174#214—44no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—42no—
180C4⋊Dic3C2×D411yescharacteristic
181Dic3⋊C4—21no—
182Dic3⋊C4—21no—
183Dic3⋊C4—42no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C4×Dic3—42no—
188C4×Dic3C2×D411yescharacteristic
189C4×Dic3C2×D411yescharacteristic
190C4×Dic3#4111yescharacteristic
191#208—42no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8C2×D411yescharacteristic
195C2×C3⋊C8C2×D411yescharacteristic
196Dic12—44no—
197D3×C8—22no—
198D3×C8—41no—
199C2×#41—31no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—62no—
203C2×SD16—62no—
204C2×M16—21no—
205C2×M16—61no—
206C22×C8—31no—
207C22×C8—31no—
208C4⋊Q8—32no—
209C4⋊Q8C22×D312yes—
210#125—62no—
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C4×Q8—31no—
214C4×Q8—31no—
215C4×Q8—62no—
216C4×Q8—62no—
217C4×D4—31no—
218C4×D4—61no—
219C4×D4—61no—
220C4×D4C22×D311yescharacteristic
221#117—31no—
222#117—61no—
223#117—62no—
224C2×C42—31no—
225C2×C42—61no—
226#107—22no—
227#107—61no—
228#107—61no—
229#106—62no—
230C4⋊C8—31no—
231C4⋊C8—32no—
232C4⋊C8—61no—
233C4⋊C8—61no—
234C4⋊C8C22×D311yescharacteristic
235C4⋊C8C22×D312yes—
236#103—62no—
237#103—62no—
238#103—62no—
239#103—62no—
240#103—62no—
241#102—22no—
242C22⋊C8—21no—
243C22⋊C8—61no—
244C22⋊C8—62no—
245C22⋊C8—61no—
246#97—61no—
247#97—61no—
248C4×C8—31no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—61no—
252C4×C8—62no—
253C4×C8C22×D311yescharacteristic
254C23×C3—21no—
255C22×D3#12111yescharacteristic
256C3×Q8—42no—
257C3×Q8—42no—
258C3×D4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266#67—42no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3#12111yescharacteristic
273C2×Dic3#12111yescharacteristic
274C2×Dic3#12111yescharacteristic
275C4×D3—21no—
276C4×D3—21no—
277C4×D3—22no—
278C4×D3—22no—
279Dic6—22no—
280Dic6—42no—
281Dic6—42no—
282C3×C8—21no—
283C3×C8—41no—
284C3×C8—42no—
285C3×C8#13512yes—
286C3⋊C8—21no—
287C3⋊C8—21no—
288C3⋊C8—41no—
289C3⋊C8—42no—
290#41—122no—
291C2×Q8—22no—
292C2×Q8—31no—
293C2×Q8—62no—
294C2×D4—61no—
295C2×D4D3×D411yescharacteristic
296C22×C4—21no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—61no—
302Q16—42no—
303Q16—42no—
304Q16—124no—
305SD16—124no—
306M16—42no—
307M16—122no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—62no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—121no—
316C2×C8D3×D411yescharacteristic
317C2×C8D3×D411yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—31no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—122no—
326C4⋊C4—122no—
327C4⋊C4D3×D411yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—122no—
332C42—31no—
333C42—31no—
334C42—31no—
335C42—61no—
336C42—61no—
337C42—61no—
338C42—122no—
339C42C23×D311yescharacteristic
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3C2×#12111yescharacteristic
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—82no—
350C3×C4C2×#13511yescharacteristic
351C3×C4#57511yescharacteristic
352C3×C4#57512yes—
353Dic3—21no—
354Dic3—21no—
355Dic3—41no—
356Dic3—41no—
357Dic3—82no—
358Dic3#47312yes—
359C23—21no—
360C23—31no—
361Q8—42no—
362Q8—42no—
363Q8—62no—
364Q8—122no—
365Q8—122no—
366D4—22no—
367D4—122no—
368C2×C4—21no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—62no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—62no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4C2×D3×D411yescharacteristic
385C2×C4C2×D3×D411yescharacteristic
386C2×C4D3×#4111yescharacteristic
387C8—21no—
388C8—41no—
389C8—42no—
390C8—61no—
391C8—61no—
392C8—121no—
393C8—122no—
394C8#91212yes—
395C2×C3—81no—
396C2×C3#303511yescharacteristic
397C2×C3#305411yescharacteristic
398C2×C3C2×#47311yescharacteristic
399D3—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—61no—
404C22—61no—
405C22—61no—
406C22D3×#12111yescharacteristic
407C4—21no—
408C4—32no—
409C4—41no—
410C4—61no—
411C4—61no—
412C4—82no—
413C4—121no—
414C4—121no—
415C4—242no—
416C4C2×#91211yescharacteristic
417C4#577311yescharacteristic
418C4#577712yes—
419C3#1916111yescharacteristic, complemented, Sylow
420C2—81no—
421C2—121no—
422C2#8033111yescharacteristic
423C2#8090311yescharacteristic
424C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1689C2
socleC22×C3C2×#121
prime radicalC2×#237D4
hypercenter#1689D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1689

Complements

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

NH ≅ G/Ntype
1. #81792C2semidirect
4. C3×#1689C2semidirect
5. #71309C2semidirect
6. #71106C2semidirect
7. #70873C2semidirect
8. #68552C2semidirect
9. #68527C2semidirect
10. #68476C2semidirect
11. #68419C2semidirect
56. #1689D3semidirect
419. C3#19161semidirect

Maximal subgroups

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

subgroupindex
#817922
#809492
#795172
C3×#16892
#713092
#711062
#708732
#685522
#685272
#684762
#684192
#191613