Finite Group Databasebeta

← #10696038

Subgroups — #10696038

Order 768.

566 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3103C211yesmaximal, characteristic
2#83315C211yesmaximal, characteristic
3#82467C211yesmaximal, characteristic, complemented
4#71816C211yesmaximal, characteristic
5#71622C211yesmaximal, characteristic, complemented
6#70770C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10014—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#481—21no—
13C3×#475—21no—
14C3×#460—21no—
15C3×#448—21no—
16C3×C4×SD16—22no—
17C3×#431C2211yescharacteristic
18C2×#837—22no—
19#5230—21no—
20#5094—21no—
21#5075—21no—
22#5040—21no—
23#5034—21no—
24C4×#214—22no—
25#4983—21no—
26#4981—21no—
27#4979—21no—
28#4932C2211yescharacteristic
29C4×Dic3⋊C4—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4535—22no—
33#4456—22no—
34#3103—31no—
35#3103D311yescharacteristic
36#2399—31no—
37#1963—31no—
38#1487—31no—
39#1487—31no—
40#1274—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×SD16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#214—22no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#837—44no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#810—22no—
86#808—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532C2×D311yescharacteristic
92C2×C4×Q8—31no—
93C2×C4×Q8C2×D311yescharacteristic
94#481—21no—
95#481—61no—
96#475—21no—
97#475—61no—
98#460—21no—
99#460—61no—
100#448—21no—
101#448—61no—
102C4×SD16—22no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423—31no—
107C2×#104—62no—
108#398—61no—
109#398—61no—
110#395—61no—
111#395—61no—
112C4×C4⋊C4—62no—
113#376—31no—
114Q8⋊C8—62no—
115Q8⋊C8—62no—
116C3×#41—42no—
117C3×#41—42no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—42no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—42no—
138C3×SD16—44no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#214—44no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—44no—
158Dic3⋊C4—44no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—22no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164C4×Dic3SD1614yescomplemented
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×#41—21no—
169C22×Q8—31no—
170C22×Q8#6711yescharacteristic
171C2×SD16—22no—
172C2×SD16—62no—
173C2×M16—61no—
174C4⋊Q8#6711yescharacteristic
175#127#6711yescharacteristic
176#124—21no—
177#121—21no—
178#121—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×D4—21no—
188C4×D4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C42—21no—
196C2×C42—32no—
197C2×C42—61no—
198C2×C42C22×D311yescharacteristic
199#106—22no—
200#106—62no—
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#104—124no—
206#103—22no—
207#103—62no—
208#102—22no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—32no—
213C4×C8#6712yes—
214#95—62no—
215#95—62no—
216#95—61no—
217#95—61no—
218C23×C3—41no—
219C23×C3#12011yescharacteristic
220C3×Q8—22no—
221C3×Q8—22no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×Q8—42no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—82no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×SD1612yes—
246C2×C3×C4#13712yes—
247C2×Dic3—22no—
248C2×Dic3—24no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255Dic6—22no—
256Dic6—22no—
257Dic6—41no—
258Dic6—41no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262#41—42no—
263#41—42no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—42no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—122no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284SD16—44no—
285SD16—124no—
286M16—122no—
287C2×C8—22no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—124no—
299C4⋊C4—124no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42—34no—
305C42—41no—
306C42—41no—
307C42—62no—
308C42—121no—
309C42—121no—
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C42C2×#6711yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#39211yescharacteristic
320C22×C3#44812yes—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#46112yes—
329C3×C4#49211yescharacteristic
330C3×C4#49711yescharacteristic
331Dic3—44no—
332Dic3—42no—
333Dic3—81no—
334C23—41no—
335C23#93711yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—41no—
339Q8—41no—
340Q8—42no—
341Q8—62no—
342Q8—62no—
343Q8—121no—
344Q8—121no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—82no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—62no—
358C2×C4—64no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4D3×SD1612yes—
368C2×C4#91512yes—
369C2×C4#93711yescharacteristic
370C2×C4#93811yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#93911yescharacteristic
373C2×C4#94011yescharacteristic
374C2×C4#95311yescharacteristic
375C8—42no—
376C8—122no—
377C8—241no—
378C2×C3—22no—
379C2×C3—161no—
380C2×C3#201011yescharacteristic
381C2×C3#201011yescharacteristic
382C2×C3#201311yescharacteristic
383C22—21no—
384C22—21no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22#517012yes—
389C22#522311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—124no—
398C4—122no—
399C4—241no—
400C4#516212yes—
401C4#516711yescharacteristic
402C4#517511yescharacteristic
403C3#1001411yescharacteristic, complemented, Sylow
404C2—22no—
405C2—161no—
406C2#7171511yescharacteristic
407C2#7205711yescharacteristic
408C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3103C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3103D3
hypocenterC3#10014
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
3. #82467C2semidirect
5. #71622C2semidirect
6. #70770C2semidirect
7. #67737C2semidirect
164. C4×Dic3SD16semidirect
403. C3#10014semidirect

Maximal subgroups

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

subgroupindex
C3×#31032
#833152
#824672
#718162
#716222
#707702
#677372
#100143