Finite Group Databasebeta

← #10697607

Subgroups — #10697607

Order 768.

556 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#3077C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#82382C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10247—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#936—21no—
12C2×#869C2211yescharacteristic
13C3×#478—22no—
14C3×#476—21no—
15C3×#474—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#418—21no—
19#5212—22no—
20#5080—21no—
21#5079—21no—
22#5043—22no—
23#5020—22no—
24#4973—21no—
25#4953—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4538—22no—
30D4⋊(C3⋊C8)—22no—
31#4482—22no—
32#3229—31no—
33#3077D311yescharacteristic
34#2547—31no—
35#1546—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#217—24no—
59#936—42no—
60#936—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D412yes—
64C2×C4×Dic3—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69C3×#107—22no—
70C3×#106—22no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#102—22no—
74C3×#102—22no—
75#835—24no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×#124—61no—
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#478—22no—
93#476—21no—
94#475—62no—
95#474—21no—
96#431—31no—
97#431—31no—
98#428—21no—
99#423C2×D311yescharacteristic
100#418—21no—
101#397—62no—
102#391—61no—
103#380—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#328—62no—
107D4⋊C8—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4SD1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—44no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—22no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—48no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—42no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C22×Q8—61no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×M16—21no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172#124—122no—
173#124—122no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182#117—21no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—64no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—22no—
192#106—22no—
193#106—64no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8#6712yes—
198#103—64no—
199#102—22no—
200#102—22no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C4×C8—61no—
204C4×C8—61no—
205#95—32no—
206#95—64no—
207C23×C3—22no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3#12011yescharacteristic
213C3×Q8—42no—
214C3×Q8—81no—
215C3×D4—22no—
216C3×D4—22no—
217C3×D4—41no—
218C3×D4—41no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4#21512yes—
265C22×C4—21no—
266C22×C4—44no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274M16—41no—
275M16—41no—
276C2×C8—22no—
277C2×C8—62no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—122no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—128no—
296C42—41no—
297C42—121no—
298C42—121no—
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C42C2×#6711yescharacteristic
302C42C2×#6711yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—82no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#39211yescharacteristic
311C22×C3#45011yescharacteristic
312C22×C3#45211yescharacteristic
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—81no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4#44811yescharacteristic
320C3×C4#44911yescharacteristic
321C3×C4#45412yes—
322C3×C4#46511yescharacteristic
323C3×C4#46811yescharacteristic
324Dic3—82no—
325Dic3—82no—
326C23—22no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23#93711yescharacteristic
332Q8—42no—
333Q8—81no—
334Q8—241no—
335Q8—241no—
336Q8—241no—
337Q8—241no—
338D4—22no—
339D4—22no—
340D4—41no—
341D4—41no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4#93711yescharacteristic
357C2×C4#93811yescharacteristic
358C2×C4#93911yescharacteristic
359C2×C4#93911yescharacteristic
360C2×C4#94011yescharacteristic
361C2×C4C2×#21511yescharacteristic
362C2×C4#94211yescharacteristic
363C2×C4#94911yescharacteristic
364C2×C4#95011yescharacteristic
365C2×C4#95311yescharacteristic
366C8—42no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—82no—
372C2×C3#200011yescharacteristic
373C2×C3#200311yescharacteristic
374C2×C3#200911yescharacteristic
375C22—21no—
376C22—21no—
377C22—44no—
378C22—42no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22#504611yescharacteristic
383C22#504711yescharacteristic
384C22#522311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—242no—
392C4—242no—
393C4#504811yescharacteristic
394C4#504911yescharacteristic
395C4#520112yes—
396C4#522711yescharacteristic
397C4#523811yescharacteristic
398C3#1024711yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—82no—
402C2#7169911yescharacteristic
403C2#7209611yescharacteristic
404C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3077D3
hypocenterC3#10247
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #83323C2semidirect
4. #71835C2semidirect
6. #70506C2semidirect
7. #67715C2semidirect
398. C3#10247semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833232
#823822
#718352
#715972
#705062
#677152
#102473