Finite Group Databasebeta

← #10696055

Subgroups — #10696055

Order 768.

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

normal subgroups only

This group has 528 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×#3104C211yesmaximal, characteristic
2#83368C211yesmaximal, characteristic
3#82467C211yesmaximal, characteristic
4#71840C211yesmaximal, characteristic
5#71622C211yesmaximal, characteristic
6#70770C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic
8#10017—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#484—21no—
13C3×#477—21no—
14C3×#462—21no—
15C3×#451—21no—
16C3×C4×Q16—22no—
17C3×#431C2211yescharacteristic
18#5230—21no—
19#5091—21no—
20#5090—21no—
21#5050—21no—
22#5044—21no—
23C4×#216—22no—
24#4983—21no—
25#4981—21no—
26#4979—21no—
27#4932C2211yescharacteristic
28C4×Dic3⋊C4—22no—
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32#4532—22no—
33#4456—22no—
34#3104—31no—
35#3104D311yescharacteristic
36#2399—31no—
37#1995—31no—
38#1487—31no—
39#1487—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×Q16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#216—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×#107—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#103—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#835—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#810—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91C2×C4×Q8—31no—
92C2×C4×Q8C2×D311yescharacteristic
93#484—21no—
94#484—61no—
95#477—21no—
96#477—61no—
97#462—21no—
98#462—61no—
99#451—21no—
100#451—61no—
101C4×Q16—22no—
102C4×Q16—62no—
103#431—31no—
104#431C2×D311yescharacteristic
105#423—31no—
106#398—61no—
107#398—61no—
108#395—61no—
109#395—61no—
110C4×C4⋊C4—62no—
111#376—31no—
112#339—62no—
113Q8⋊C8—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—42no—
131C3×Q16—44no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—44no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—44no—
155Dic3⋊C4—44no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—22no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3Q1614yescomplemented
162#208—42no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C22×Q8—31no—
166C22×Q8#6711yescharacteristic
167C2×Q16—22no—
168C2×Q16—62no—
169C2×M16—61no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
172#125—21no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185#117—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42—32no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#107—22no—
197#107—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#103—22no—
203#103—22no—
204#103—62no—
205#103—62no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208C4×C8—32no—
209C4×C8#6712yes—
210#95—62no—
211#95—62no—
212#95—61no—
213#95—61no—
214C23×C3#12011yescharacteristic
215C3×Q8—22no—
216C3×Q8—22no—
217C3×Q8—41no—
218C3×Q8—41no—
219C3×Q8—42no—
220C3×Q8—42no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×C3×C4C2×Q1612yes—
237C2×C3×C4#13712yes—
238C2×Dic3—22no—
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246Dic6—22no—
247Dic6—22no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—122no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—62no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4C2×#6711yescharacteristic
269Q16—44no—
270Q16—124no—
271M16—122no—
272C2×C8—22no—
273C2×C8—62no—
274C2×C8—121no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—124no—
288C4⋊C4—124no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C22⋊C4—41no—
292C42—34no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—62no—
297C42—121no—
298C42—121no—
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C42C2×#6711yescharacteristic
302C42C2×#6711yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#39211yescharacteristic
306C22×C3#45112yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4#46212yes—
316C3×C4#49411yescharacteristic
317C3×C4#49711yescharacteristic
318Dic3—44no—
319Dic3—42no—
320Dic3—81no—
321C23#93711yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—42no—
327Q8—42no—
328Q8—62no—
329Q8—62no—
330Q8—121no—
331Q8—121no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—62no—
342C2×C4—64no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4D3×Q1612yes—
350C2×C4#91812yes—
351C2×C4#93711yescharacteristic
352C2×C4#93811yescharacteristic
353C2×C4#93911yescharacteristic
354C2×C4#93911yescharacteristic
355C2×C4#94011yescharacteristic
356C2×C4#95311yescharacteristic
357C8—42no—
358C8—122no—
359C8—241no—
360C2×C3—22no—
361C2×C3#201011yescharacteristic
362C2×C3#201011yescharacteristic
363C2×C3#201611yescharacteristic
364C22—21no—
365C22—21no—
366C22#518612yes—
367C22#522311yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—124no—
377C4—122no—
378C4—241no—
379C4#518212yes—
380C4#518311yescharacteristic
381C4#519011yescharacteristic
382C3#1001711yescharacteristic, complemented, Sylow
383C2—22no—
384C2#7171511yescharacteristic
385C2#7207311yescharacteristic
386C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3104C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3104D3
hypocenterC3#10017
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
161. C4×Dic3Q16semidirect
382. C3#10017semidirect

Maximal subgroups

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

subgroupindex
C3×#31042
#833682
#824672
#718402
#716222
#707702
#677372
#100173