Finite Group Databasebeta

← #10696051

Subgroups — #10696051

Order 768.

560 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2398C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic, complemented
3#82314C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic
6#69441C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#400—21no—
13C3×#397—21no—
14C3×#394—21no—
15C3×#393—21no—
16C3×C4×C22⋊C4—22no—
17C3×#376C2211yescharacteristic
18#5228—21no—
19#5218—21no—
20#5031—21no—
21C4×#216—22no—
22#5021—21no—
23#5019—21no—
24C4×#214—22no—
25#5008—21no—
26#4971—21no—
27#4970—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32D4⋊(C3⋊C8)—22no—
33#4523—22no—
34#3104—31no—
35#3101—31no—
36#2398D311yescharacteristic
37#1995—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59C2×C3×C42D412yes—
60C2×C3×C42C2311yescharacteristic
61C2×#216—22no—
62C2×#214—22no—
63#926—21no—
64#924—21no—
65#924—21no—
66C2×#208—21no—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—21no—
75C3×#95—21no—
76#835—22no—
77#832—22no—
78#830—21no—
79#830—21no—
80#810—22no—
81#810—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4×C3⋊C8D412yes—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4C2×D311yescharacteristic
93#484—61no—
94#483—61no—
95#477—61no—
96#476—61no—
97#462—61no—
98#461—61no—
99#451—61no—
100#450—61no—
101C4×Q16—62no—
102C4×SD16—62no—
103#431—31no—
104#431—31no—
105#423—31no—
106#400—21no—
107#397—21no—
108#394—21no—
109#393—21no—
110C4×C22⋊C4—22no—
111#376C2×D311yescharacteristic
112#339—62no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C2×Dic6—21no—
134C2×Dic6—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×C22⋊C4—44no—
142C3×C22⋊C4—44no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—41no—
154#216—44no—
155#214—44no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8#6711yescharacteristic
168C22×D4#6711yescharacteristic
169C23×C4—21no—
170C2×Q16—62no—
171C2×SD16—62no—
172C2×M16—61no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175#125—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188#117—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C42—21no—
196C2×C42#6712yes—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#106—62no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C4×C8—32no—
211C4×C8—32no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—21no—
216C23×C3—22no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3#12011yescharacteristic
222C3×Q8—22no—
223C3×Q8—22no—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×D4—22no—
227C3×D4—22no—
228C3×D4—41no—
229C3×D4—41no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—82no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12711yescharacteristic
251C2×C3×C4#13512yes—
252C2×C3×C4#13712yes—
253C2×Dic3—41no—
254C2×Dic3—41no—
255Dic6—42no—
256Dic6—42no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C3⋊C8—81no—
260C24—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C2×#6711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279C22×C4C2×#6711yescharacteristic
280Q16—124no—
281SD16—124no—
282M16—122no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—121no—
303C42—22no—
304C42—22no—
305C42—41no—
306C42—41no—
307C42—121no—
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—82no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#39211yescharacteristic
320C22×C3#45212yes—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—82no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4#46412yes—
330C3×C4#49511yescharacteristic
331C3×C4#49711yescharacteristic
332Dic3—81no—
333Dic3—81no—
334C23—22no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23#95311yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—122no—
345Q8—122no—
346D4—22no—
347D4—22no—
348D4—41no—
349D4—41no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#93711yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93811yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#94011yescharacteristic
373C2×C4#95012yes—
374C2×C4#95112yes—
375C8—122no—
376C8—122no—
377C8—241no—
378C2×C3—22no—
379C2×C3—82no—
380C2×C3#201011yescharacteristic
381C2×C3#201011yescharacteristic
382C2×C3#201511yescharacteristic
383C22—21no—
384C22—21no—
385C22—44no—
386C22—42no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22#522311yescharacteristic
391C22#523912yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—241no—
401C4—241no—
402C4#505412yes—
403C4#505911yescharacteristic
404C4#506011yescharacteristic
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—82no—
408C2#7171511yescharacteristic
409C2#7171511yescharacteristic
410C2#7172911yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2398C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2398D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. #82447C2semidirect
4. #71622C2semidirect
6. #69441C2semidirect
7. #67614C2semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
C3×#23982
#824472
#823142
#716222
#715982
#694412
#676142
#100163