Finite Group Databasebeta

← #10698939

Subgroups — #10698939

Order 768.

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

normal subgroups only

This group has 420 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×#3037C211yesmaximal, characteristic
2#71814C211yesmaximal, characteristic
3#71804C211yesmaximal, characteristic
4#71627C211yesmaximal, characteristic, complemented
5#71624C211yesmaximal, characteristic, complemented
6#70493C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#10467—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#464—22no—
13C3×#462—21no—
14C3×#460—21no—
15C3×#423C2211yescharacteristic
16C3×#417—21no—
17C2×#835—21no—
18#4936C2211yescharacteristic
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4547—21no—
23#4544—21no—
24#4537—22no—
25#4529—21no—
26#4528—22no—
27#4480—22no—
28#3037D311yescharacteristic
29#1805—31no—
30#1570—31no—
31#1540—31no—
32#1517—31no—
33#1461—31no—
34#1286—31no—
35C2×C3×#41—21no—
36C22×C3×Q8D411yescharacteristic
37C3×#137—42no—
38C3×#137—42no—
39C2×C3×Q16—22no—
40C2×C3×SD16—22no—
41C2×C3×M16—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D412yes—
61C3×#103—22no—
62C3×#102—22no—
63#835—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#830D412yes—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C2×#137—21no—
75#532C2×D311yescharacteristic
76C2×C4×Q8C2×D311yescharacteristic
77#464—22no—
78#462—21no—
79#460—21no—
80#431—31no—
81#431—31no—
82#423C2×D311yescharacteristic
83C2×C4⋊C8—31no—
84#417—21no—
85C2×#103—61no—
86#382—31no—
87#340—61no—
88#334—62no—
89#332—61no—
90#331—61no—
91#327—62no—
92#324—62no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8SD1612yes—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—41no—
115C3×Q16—44no—
116C3×SD16—44no—
117C3×M16—41no—
118C3×M16—41no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×#41—21no—
142C22×Q8#6711yescharacteristic
143#137—42no—
144#137—42no—
145C2×Q16—22no—
146C2×SD16—22no—
147C2×M16—21no—
148C22×C8—61no—
149C4⋊Q8#6711yescharacteristic
150#127#6711yescharacteristic
151#124—21no—
152#121—21no—
153#121—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×D4—21no—
159C4×D4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4#6711yescharacteristic
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165C4⋊C8—21no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8#6712yes—
171#103—22no—
172#103—122no—
173#102—22no—
174C22⋊C8—32no—
175C22⋊C8—61no—
176C4×C8—61no—
177C4×C8—61no—
178#95—61no—
179#95—61no—
180C23×C3—41no—
181C23×C3#12111yescharacteristic
182C3×Q8—22no—
183C3×Q8—22no—
184C3×Q8—41no—
185C3×Q8—41no—
186C3×Q8—41no—
187C3×Q8—41no—
188C3×D4—41no—
189C3×D4—41no—
190C3×D4—81no—
191C3×D4—81no—
192C3×D4—81no—
193C3×D4—81no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4#12311yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×C3×C4C2×SD1611yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C3⋊C8—81no—
221#41—41no—
222#41—41no—
223#41—41no—
224#41—41no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8#21412yes—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—121no—
240C22×C4D3×D411yescharacteristic
241C22×C4#23711yescharacteristic
242C22×C4C2×#6711yescharacteristic
243Q16—44no—
244SD16—44no—
245M16—41no—
246M16—41no—
247C2×C8—22no—
248C2×C8—62no—
249C2×C8—121no—
250C2×C8—121no—
251C2×C8—121no—
252C2×C8—121no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42—41no—
265C42#23711yescharacteristic
266C42#23711yescharacteristic
267C42C2×#6711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3#39411yescharacteristic
275C22×C3#46511yescharacteristic
276C22×C3#46911yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—81no—
283C3×C4—81no—
284C3×C4#44911yescharacteristic
285C3×C4#45011yescharacteristic
286C3×C4#48111yescharacteristic
287C3×C4#48211yescharacteristic
288Dic3—161no—
289C23—41no—
290C23#93911yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297D4—41no—
298D4—41no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#93211yescharacteristic
317C2×C4C2×#21411yescharacteristic
318C2×C4#93511yescharacteristic
319C2×C4#93511yescharacteristic
320C2×C4#93611yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#94911yescharacteristic
324C2×C4#95011yescharacteristic
325C2×C4#95311yescharacteristic
326C8—42no—
327C8—122no—
328C8—241no—
329C8—241no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—161no—
333C2×C3#203411yescharacteristic
334C2×C3#204911yescharacteristic
335C2×C3#205311yescharacteristic
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#503911yescharacteristic
342C22#504111yescharacteristic
343C22#522211yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—481no—
352C4#503411yescharacteristic
353C4#503511yescharacteristic
354C4#521811yescharacteristic
355C4#523811yescharacteristic
356C3#1046711yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—161no—
360C2#7168011yescharacteristic
361C2#7210411yescharacteristic
362C2#7211511yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#394
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10467
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
4. #71627C2semidirect
5. #71624C2semidirect
6. #70493C2semidirect
7. #67713C2semidirect
356. C3#10467semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#718142
#718042
#716272
#716242
#704932
#677132
#104673