Finite Group Databasebeta

← #10696052

Subgroups — #10696052

Order 768.

560 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 95 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#82492C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic, complemented
3#79298C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70515C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5231—21no—
15#5228—21no—
16#5031—21no—
17C4×#216—22no—
18#5021—21no—
19#4992—21no—
20#4991—21no—
21#4986—21no—
22#4970—21no—
23C4×(C2×D3)⋊C4—22no—
24#4932C2211yescharacteristic
25#4925C2211yescharacteristic
26#4842—21no—
27#4841—21no—
28#4731—21no—
29#4721—21no—
30C4×#204—22no—
31C3×Q8⋊C8—22no—
32#4555—22no—
33#4483—22no—
34#3104—31no—
35#3101—31no—
36#2398—31no—
37#1995—31no—
38#1487D311yescharacteristic, complemented
39#1486—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57#926—21no—
58C2×C4⋊Dic3D411yescharacteristic
59#924—21no—
60#924—21no—
61C2×C4×Dic3D412yes—
62C2×#204—22no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#821—22no—
84#817—22no—
85#816—22no—
86#810—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4—31no—
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#431C2×D311yescharacteristic
105#423C2×D311yescharacteristic
106#400—61no—
107#397—61no—
108#394—61no—
109#393—61no—
110C4×C22⋊C4—62no—
111#376—31no—
112#339—62no—
113Q8⋊C8—22no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—42no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—42no—
132C2×C4×D3—42no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#217—41no—
149#216—44no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—44no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—22no—
163C4×Dic3—22no—
164C4×Dic3—41no—
165C2×C3⋊C8—22no—
166#204—44no—
167C22×Q8D1211yescharacteristic
168C22×D4—31no—
169C23×C4—61no—
170C2×Q16—62no—
171C2×SD16—62no—
172C2×M16—21no—
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—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—32no—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C4×C8—32no—
211C4×C8#6712yes—
212#95—62no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3#12011yescharacteristic
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C3×Q8—22no—
223C3×Q8—22no—
224C3×Q8—41no—
225C3×Q8—41no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×C3×C4#13512yes—
240C2×C3×C4#13712yes—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248D12—22no—
249D12—22no—
250D12—41no—
251D12—41no—
252C4×D3—82no—
253C4×D3—81no—
254C4×D3—81no—
255Dic6—42no—
256Dic6—42no—
257C3×C8—42no—
258C3×C8—81no—
259C3⋊C8—42no—
260C24—61no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—122no—
275C22×C4—122no—
276C22×C4—121no—
277C22×C4C2×D1211yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280Q16—124no—
281SD16—124no—
282M16—42no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—41no—
304C42—41no—
305C42—62no—
306C42—62no—
307C42—121no—
308C42#23511yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#39211yescharacteristic
315C22×C3#45212yes—
316C2×D3—44no—
317C2×D3—42no—
318C2×D3—82no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—81no—
327C3×C4#46412yes—
328C3×C4#49511yescharacteristic
329C3×C4#49711yescharacteristic
330Dic3—82no—
331Dic3—81no—
332Dic3—81no—
333Dic3—81no—
334C23—62no—
335C23—122no—
336C23—121no—
337C23—121no—
338C23—122no—
339C23#88211yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—122no—
345Q8—122no—
346D4—62no—
347D4—62no—
348D4—121no—
349D4—121no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—242no—
367C2×C4#88211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#88411yescharacteristic
370C2×C4#91612yes—
371C2×C4#91812yes—
372C2×C4#93011yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#94011yescharacteristic
375C8—42no—
376C8—81no—
377C8—122no—
378C2×C3—22no—
379C2×C3#201011yescharacteristic
380C2×C3#201011yescharacteristic
381C2×C3#201511yescharacteristic
382D3—82no—
383C22—21no—
384C22—21no—
385C22—124no—
386C22—122no—
387C22—242no—
388C22—241no—
389C22—241no—
390C22#480812yes—
391C22#496011yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—242no—
399C4—241no—
400C4—241no—
401C4—241no—
402C4#479812yes—
403C4#516811yescharacteristic
404C4#519011yescharacteristic
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—242no—
408C2#6999311yescharacteristic
409C2#7027811yescharacteristic
410C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1487D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. #82447C2semidirect
4. C3×#1487C2semidirect
5. #70515C2semidirect
7. #67737C2semidirect
38. #1487D3semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
#824922
#824472
#792982
C3×#14872
#705152
#704962
#677372
#100163