Finite Group Databasebeta

← #10697625

Subgroups — #10697625

Order 768.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82384C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic, complemented
6#70780C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10249—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5221—21no—
15#5215—21no—
16#5212—22no—
17#5009—22no—
18#5008—21no—
19#5007—21no—
20#4977—21no—
21#4964—21no—
22#4947—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4850—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29C3×D4⋊C8—22no—
30#4482—22no—
31#4467—22no—
32#3230—31no—
33#3077—31no—
34#2563—31no—
35#1549—31no—
36#1503—31no—
37#1478D311yescharacteristic
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—24no—
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54#924—21no—
55#924—21no—
56C2×C4×Dic3—21no—
57C2×#208—21no—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73#832—22no—
74#832—22no—
75C3⋊#95—24no—
76C3⋊#95D412yes—
77#818—24no—
78#816—24no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×D4C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—61no—
93#431C2×D311yescharacteristic
94#431C2×D311yescharacteristic
95#428—61no—
96#423—31no—
97#418—61no—
98#393—62no—
99#390—61no—
100#382—61no—
101#380—62no—
102C4×C4⋊C4—31no—
103#332—62no—
104#328—62no—
105D4⋊C8—22no—
106C24×C3—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4Q1612yes—
113C22×C3×C4—21no—
114C22×C3×C4—44no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—22no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—82no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138#217—84no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—42no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C22×Q8—61no—
157C22×D4D1211yescharacteristic
158C23×C4—21no—
159C2×M16—61no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175#117—61no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—64no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—64no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—64no—
192#102—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C4×C8—21no—
197C4×C8—21no—
198#95—32no—
199#95—64no—
200C23×C3—22no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3#12011yescharacteristic
206C3×D4—22no—
207C3×D4—22no—
208C3×D4—41no—
209C3×D4—41no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—82no—
215C2×C3×C4—82no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×Q1611yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—44no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233Dic6—42no—
234Dic6—82no—
235Dic6—82no—
236Dic6—81no—
237C3×C8—42no—
238C3×C8—81no—
239C3⋊C8—42no—
240C24—21no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4Dic1212yes—
252C22×C4—21no—
253C22×C4—44no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4C2×D1211yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261M16—121no—
262M16—121no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—62no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—61no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—122no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—121no—
282C22⋊C4—244no—
283C42—121no—
284C42—121no—
285C42—121no—
286C42#23511yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C42C2×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—44no—
293C22×C3—42no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C22×C3#45111yescharacteristic
299C22×C3#45211yescharacteristic
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—81no—
304C3×C4#45111yescharacteristic
305C3×C4#45211yescharacteristic
306C3×C4#45412yes—
307C3×C4#46711yescharacteristic
308C3×C4#47011yescharacteristic
309Dic3—82no—
310Dic3—81no—
311Dic3—82no—
312Dic3—81no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23#88211yescharacteristic
319Q8—122no—
320Q8—241no—
321Q8—242no—
322Q8—242no—
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—122no—
334C2×C4—124no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4#88211yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#90411yescharacteristic
344C2×C4C2×Dic1211yescharacteristic
345C2×C4#90911yescharacteristic
346C2×C4#90911yescharacteristic
347C2×C4#93011yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#94011yescharacteristic
350C8—42no—
351C8—81no—
352C8—122no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—82no—
356C2×C3#200111yescharacteristic
357C2×C3#200311yescharacteristic
358C2×C3#200811yescharacteristic
359C22—21no—
360C22—21no—
361C22—44no—
362C22—42no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22#473311yescharacteristic
367C22#473911yescharacteristic
368C22#496011yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—81no—
373C4—242no—
374C4—242no—
375C4—241no—
376C4—241no—
377C4#473311yescharacteristic
378C4#473911yescharacteristic
379C4#511611yescharacteristic
380C4#513711yescharacteristic
381C4#520112yes—
382C3#1024911yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—82no—
386C2#6996411yescharacteristic
387C2#7032611yescharacteristic
388C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1478D3
hypocenterC3#10249
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #79272C2semidirect
5. #70792C2semidirect
6. #70780C2semidirect
7. #67746C2semidirect
382. C3#10249semidirect

Maximal subgroups

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

subgroupindex
#823842
#822962
#792722
C3×#14782
#707922
#707802
#677462
#102493