Finite Group Databasebeta

← #10696757

Subgroups — #10696757

Order 768.

544 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3153C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic, complemented
3#82949C211yesmaximal, characteristic, complemented
4#71803C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#67670C211yesmaximal, characteristic, complemented
8#10121—31nomaximal, Sylow
9C2×C3×#137—21no—
10C2×C3×AGL(1,ℤ/8)—21no—
11C3×#532C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#497—21no—
15C3×#496—21no—
16C3×#495—22no—
17C3×C4×M16C2211yescharacteristic
18#5228—21no—
19#5218—21no—
20#5081—21no—
21#5071—21no—
22#5060—21no—
23#5055—21no—
24#5054—22no—
25#5035—22no—
26#5008—21no—
27#5007—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4525—22no—
33#4456—22no—
34#3235—31no—
35#3215—31no—
36#3153D311yescharacteristic
37#1496—31no—
38#1485—31no—
39#1469—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C3×#137—44no—
45C3×AGL(1,ℤ/8)—44no—
46C2×C3×Q16—22no—
47C2×C3×SD16—22no—
48C2×C3×SD16—22no—
49C2×C3×D8—22no—
50C2×C3×M16—21no—
51C3×C4⋊Q8D411yescharacteristic
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#124—21no—
55C3×#124—21no—
56C3×#124—21no—
57C3×#124—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#216—22no—
66C2×#214—22no—
67#926—21no—
68#924—21no—
69#924—21no—
70#870—21no—
71#869D411yescharacteristic
72#869D411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×#97—21no—
76C3×C4×C8D412yes—
77#835—22no—
78#832—22no—
79#832—22no—
80#832—22no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89#803—21no—
90#803—21no—
91C4×C3⋊C8—21no—
92C2×#137—21no—
93C2×AGL(1,ℤ/8)—21no—
94#533—31no—
95#532C2×D311yescharacteristic
96C2×#124C2×D311yescharacteristic
97#501—61no—
98#497—21no—
99#497—61no—
100#496—21no—
101#495—22no—
102#490—61no—
103#489—61no—
104#482—62no—
105#476—61no—
106#474—61no—
107#464—62no—
108#451—61no—
109#450—61no—
110#432—31no—
111#432—31no—
112#431—31no—
113C4×M16C2×D311yescharacteristic
114Q8⋊C8—62no—
115D4⋊C8—62no—
116#323—62no—
117C24×C3—21no—
118C3×#41—42no—
119C3×#41—42no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—42no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—42no—
130C2×C3×D4—41no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×Dic6—21no—
137C2×Dic6—21no—
138C3×Q16—44no—
139C3×SD16—44no—
140C3×SD16—44no—
141C3×D8—44no—
142C3×M16—24no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156#217—41no—
157#216—82no—
158#214—82no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×#41—21no—
170C22×Q8#6711yescharacteristic
171C22×D4#6711yescharacteristic
172#137—44no—
173AGL(1,ℤ/8)—44no—
174C2×Q16—22no—
175C2×Q16—62no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—62no—
179C2×D8—22no—
180C2×M16—21no—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183C4⋊Q8#6711yescharacteristic
184#127#6711yescharacteristic
185#125—61no—
186#124—21no—
187#124—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191C22⋊Q8—61no—
192C22⋊Q8—61no—
193#121—21no—
194#121—21no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—21no—
198C4×D4—21no—
199#117—61no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42C22×D311yescharacteristic
203#107—62no—
204#106—62no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—62no—
209#102—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#97—21no—
216#97—61no—
217#97—61no—
218C4×C8—61no—
219C4×C8#6712yes—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3#12111yescharacteristic
226C3×Q8—41no—
227C3×Q8—42no—
228C3×Q8—41no—
229C3×Q8—42no—
230C3×D4—41no—
231C3×D4—42no—
232C3×D4—41no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—82no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—82no—
243C2×C3×C4—81no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×Dic3—41no—
251C2×Dic3—41no—
252Dic6—42no—
253Dic6—42no—
254C3×C8—24no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259#41—42no—
260#41—42no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—44no—
280Q16—242no—
281SD16—44no—
282SD16—44no—
283SD16—242no—
284D8—44no—
285M16—24no—
286C2×C8—22no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—121no—
305C42—121no—
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42C2×#6711yescharacteristic
309C42C2×#6711yescharacteristic
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—162no—
320C22×C3#39211yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#45312yes—
329Dic3—81no—
330Dic3—81no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—81no—
335C23—81no—
336C23#93911yescharacteristic
337Q8—41no—
338Q8—42no—
339Q8—41no—
340Q8—42no—
341Q8—122no—
342Q8—122no—
343D4—41no—
344D4—42no—
345D4—41no—
346D4—42no—
347D4—42no—
348D4—42no—
349D4—82no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4#93711yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#93811yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4#94011yescharacteristic
364C2×C4#95311yescharacteristic
365C8—24no—
366C8—122no—
367C8—122no—
368C8—241no—
369C2×C3—41no—
370C2×C3—161no—
371C2×C3—161no—
372C2×C3#200211yescharacteristic
373C2×C3#200911yescharacteristic
374C2×C3#201711yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—162no—
385C22#522311yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—81no—
392C4—81no—
393C4—241no—
394C4—241no—
395C4#507712yes—
396C3#1012111yescharacteristic, complemented, Sylow
397C2—41no—
398C2—161no—
399C2—161no—
400C2#7176911yescharacteristic
401C2#7212011yescharacteristic
402C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3153C2
socleC22×C3#392
hypercenter#3153D3
hypocenterC3#10121
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
2. #83274C2semidirect
3. #82949C2semidirect
4. #71803C2semidirect
5. #71720C2semidirect
6. #69856C2semidirect
7. #67670C2semidirect
396. C3#10121semidirect

Maximal subgroups

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

subgroupindex
C3×#31532
#832742
#829492
#718032
#717202
#698562
#676702
#101213