Finite Group Databasebeta

← #10692805

Subgroups — #10692805

Order 768.

490 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 89 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#78527C211yesmaximal, characteristic
2#73395C211yesmaximal, characteristic
3#70836C211yesmaximal, characteristic, complemented
4#70808C211yesmaximal, characteristic
5#69433C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9439—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5131—21no—
15#5120—21no—
16#4940—21no—
17C4×(C2×D3)⋊C4—21no—
18#4932C2211yescharacteristic
19#4926C2211yescharacteristic
20#4714C412yescomplemented
21#4714C412yescomplemented
22#4697—22no—
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4646—21no—
27#4467—21no—
28#4467—21no—
29#4464—22no—
30#4452—21no—
31#4452—21no—
32#3140—31no—
33#2372—31no—
34#1533—31no—
35#1506—31no—
36#1506—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50#926—21no—
51#924—21no—
52#924—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56#875—21no—
57#875—21no—
58#875C2×C412yes—
59#875D412yes—
60#870—21no—
61#869C2×C411yescharacteristic, complemented
62#869C2×C411yescharacteristic, complemented
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#97C2×C412yescomplemented
69C3×C4×C8—21no—
70C3×#95—21no—
71C3×#95D411yescharacteristic
72C3×#95D411yescharacteristic
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95D411yescharacteristic
77C3⋊#95D411yescharacteristic
78#830—21no—
79#821—22no—
80#821—22no—
81#816—22no—
82#816—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#533—31no—
86C2×#124—31no—
87#489—32no—
88#489—32no—
89#488—62no—
90#432—31no—
91#432C2×D311yescharacteristic
92#419—61no—
93#418—61no—
94C4×M16C2×D311yescharacteristic
95#391—61no—
96#388—61no—
97#388—61no—
98#386—61no—
99C4×C22⋊C4—61no—
100C4×C22⋊C4—61no—
101#376—31no—
102#376C2×D311yescharacteristic
103#329—62no—
104#328—61no—
105#328—61no—
106#328—61no—
107#328—61no—
108C23×D3—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×D12—21no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12C22⋊C412yes—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×C4×D3—81no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6C22⋊C412yes—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C42—42no—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—42no—
145(C2×D3)⋊C4—42no—
146(C2×D3)⋊C4—81no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—81no—
154C4×Dic3—42no—
155C4×Dic3—41no—
156C4×Dic3—81no—
157C2×C3⋊C8—41no—
158C22×Q8—31no—
159C22×D4—31no—
160C23×C4—61no—
161C2×M16—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165#124—32no—
166#124—32no—
167#124—61no—
168#124—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C42—21no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—61no—
186#103—62no—
187#103—62no—
188#102—62no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192#97—21no—
193#97—61no—
194#97C4×D312yescomplemented
195C4×C8—21no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95D1211yescharacteristic
203#95D1211yescharacteristic
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239D12—42no—
240D12—81no—
241C4×D3—84no—
242Dic6—42no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—81no—
249C24—61no—
250C2×Q8—32no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4—121no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—122no—
266C22×C4—122no—
267C22×C4—241no—
268C22×C4C2×D1211yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—121no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—241no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—122no—
288C22⋊C4—122no—
289C22⋊C4—121no—
290C22⋊C4—241no—
291C42—42no—
292C42—122no—
293C42—121no—
294C42—241no—
295C42C2×C4×D311yescharacteristic
296C42#23511yescharacteristic
297C42D3×D411yescharacteristic
298C42#23911yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3#38611yescharacteristic
302C22×C3#45511yescharacteristic
303C22×C3#45511yescharacteristic
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—42no—
318C3×C4—81no—
319Dic3—42no—
320Dic3—42no—
321Dic3—81no—
322Dic3—81no—
323Dic3—81no—
324C23—62no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23#88211yescharacteristic
332Q8—122no—
333Q8—241no—
334Q8—241no—
335Q8—241no—
336D4—122no—
337D4—241no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—244no—
360C2×C4C4×D1211yescharacteristic
361C2×C4D3×C22⋊C411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#89311yescharacteristic
364C2×C4#89411yescharacteristic
365C2×C4#89511yescharacteristic
366C8—42no—
367C8—81no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#188511yescharacteristic
372C2×C3#190611yescharacteristic
373C2×C3#190611yescharacteristic
374D3—82no—
375C22—21no—
376C22—21no—
377C22—122no—
378C22—122no—
379C22—122no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#466511yescharacteristic
385C22#482511yescharacteristic
386C22#482511yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—122no—
396C4—122no—
397C4—241no—
398C4—241no—
399C4—241no—
400C3#943911yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—242no—
404C2#6875911yescharacteristic
405C2#6877811yescharacteristic
406C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9439
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #70836C2semidirect
5. #69433C2semidirect
6. C3×#1300C2semidirect
7. #67665C2semidirect
20. #4714C4semidirect
21. #4714C4semidirect
38. #1300D3semidirect
61. #869C2×C4semidirect
62. #869C2×C4semidirect
68. C3×#97C2×C4semidirect
194. #97C4×D3semidirect
400. C3#9439semidirect

Maximal subgroups

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

subgroupindex
#785272
#733952
#708362
#708082
#694332
C3×#13002
#676652
#94393