Finite Group Databasebeta

← #10696130

Subgroups — #10696130

Order 768.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#83316C211yesmaximal, characteristic, complemented
2#83310C211yesmaximal, characteristic
3#79355C211yesmaximal, characteristic, complemented
4C3×#1530C211yesmaximal, characteristic, complemented
5#69838C211yesmaximal, characteristic, complemented
6#69798C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#10028—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14#5091—21no—
15#5087—21no—
16#5075—21no—
17#5072—21no—
18#5040—21no—
19#5038—22no—
20#5033—21no—
21#5018—22no—
22#5015—21no—
23#5007—21no—
24#4926C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4850—21no—
27#4848—21no—
28#4738—21no—
29#4736—21no—
30#4710—22no—
31C3×#329—22no—
32#4550—22no—
33#4464—22no—
34#3245—31no—
35#3112—31no—
36#3112—31no—
37#1985—31no—
38#1530—31no—
39#1530D311yescharacteristic, complemented
40#1290—31no—
41C2×C3×#41—21no—
42C2×#235—21no—
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C22×C3⋊C8D412yes—
61C2×#204—22no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97—21no—
75C3×#97D412yes—
76#832—22no—
77#832—22no—
78#830—21no—
79#821—22no—
80#818—22no—
81#816—22no—
82#810—22no—
83#809—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89#803D412yes—
90#532—31no—
91#532C2×D311yescharacteristic
92C2×C4×Q8—31no—
93#480—61no—
94#480—61no—
95#477—61no—
96#477—61no—
97#474—61no—
98#474—61no—
99#466—62no—
100#460—61no—
101#460—61no—
102#459—61no—
103#459—61no—
104#448—61no—
105#448—61no—
106#440—62no—
107#440—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110C2×C4⋊C8—31no—
111C2×#97C2×D311yescharacteristic
112#336—62no—
113#329—22no—
114#329—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128#235—82no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—42no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149#214—44no—
150#213—82no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—42no—
164#204—44no—
165C2×#41—21no—
166C2×#41—61no—
167C22×Q8—31no—
168C2×SD16—62no—
169C2×SD16—62no—
170C2×D8—62no—
171C22×C8—21no—
172C22×C8—32no—
173C4⋊Q8—31no—
174C4⋊Q8D1211yescharacteristic
175#127—31no—
176#127D1211yescharacteristic
177#124—21no—
178#124—61no—
179#121—21no—
180#121—21no—
181#121—61no—
182#121—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—61no—
190C4×D4—61no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#107—62no—
197#107—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#103—62no—
203#103—62no—
204#102—62no—
205#102—62no—
206#102—62no—
207#102—62no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210#97—21no—
211#97—21no—
212#97—32no—
213#97#6712yes—
214C23×C3—41no—
215C23×C3#12111yescharacteristic
216C22×D3—41no—
217C3×Q8—81no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—81no—
222C3×D4—82no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—82no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12711yescharacteristic
235#67—82no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242D12—42no—
243D12—41no—
244D12—41no—
245D12—81no—
246C4×D3—82no—
247Dic6—22no—
248Dic6—22no—
249Dic6—41no—
250Dic6—41no—
251Dic6—81no—
252C3×C8—42no—
253C3×C8—42no—
254C3⋊C8—42no—
255C3⋊C8—42no—
256#41—82no—
257#41—242no—
258C2×Q8—21no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—122no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—121no—
275C22×C4#23511yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278SD16—124no—
279SD16—124no—
280D8—242no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—122no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—121no—
301C42—121no—
302C42C2×D1211yescharacteristic
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C42C2×#6711yescharacteristic
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3#39211yescharacteristic
312C22×C3#47212yes—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—81no—
322C3×C4#45511yescharacteristic
323C3×C4#45611yescharacteristic
324Dic3—42no—
325Dic3—81no—
326Dic3—81no—
327C23—41no—
328C23—121no—
329C23#88411yescharacteristic
330Q8—62no—
331Q8—62no—
332Q8—81no—
333Q8—121no—
334Q8—121no—
335Q8—241no—
336D4—42no—
337D4—41no—
338D4—41no—
339D4—81no—
340D4—82no—
341D4—122no—
342D4—121no—
343D4—121no—
344D4—241no—
345D4—242no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—62no—
352C2×C4—82no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—242no—
359C2×C4#88211yescharacteristic
360C2×C4#88211yescharacteristic
361C2×C4#88411yescharacteristic
362C2×C4#93011yescharacteristic
363C2×C4#93711yescharacteristic
364C2×C4#94011yescharacteristic
365C8—42no—
366C8—42no—
367C8—122no—
368C8—122no—
369C2×C3—22no—
370C2×C3—161no—
371C2×C3#200711yescharacteristic
372C2×C3#201011yescharacteristic
373C2×C3#201811yescharacteristic
374D3—161no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22#474912yes—
384C22#496011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—122no—
392C4—241no—
393C4—241no—
394C4#482511yescharacteristic
395C4#482611yescharacteristic
396C3#1002811yescharacteristic, complemented, Sylow
397C2—22no—
398C2—161no—
399C2—481no—
400C2#7009011yescharacteristic
401C2#7039311yescharacteristic
402C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1530C2
socleC22×C3#392
prime radical#83310C2
hypercenter#1530D3
hypocenterC3#10028
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
1. #83316C2semidirect
3. #79355C2semidirect
4. C3×#1530C2semidirect
5. #69838C2semidirect
6. #69798C2semidirect
7. #67755C2semidirect
39. #1530D3semidirect
396. C3#10028semidirect

Maximal subgroups

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

subgroupindex
#833162
#833102
#793552
C3×#15302
#698382
#697982
#677552
#100283