Finite Group Databasebeta

← #10696727

Subgroups — #10696727

Order 768.

492 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 87 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#83321C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4C3×#1525C211yesmaximal, characteristic
5#69877C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#67669C211yesmaximal, characteristic, complemented
8#10116—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×#824—22no—
15#5121—22no—
16#5091—21no—
17#5080—21no—
18#5075—21no—
19#5070—21no—
20#5042—22no—
21#5034—22no—
22#5019—21no—
23#5008—22no—
24#5007—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4702—22no—
28#4682—21no—
29#4680—21no—
30C3×#329—22no—
31#4456—22no—
32#3254—31no—
33#3227—31no—
34#3165—31no—
35#1966—31no—
36#1525D311yescharacteristic
37#1485—31no—
38#1299—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#214—22no—
52#926—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D412yes—
55#924—21no—
56#924—21no—
57C2×#208D412yes—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#97—21no—
68C3×#97D412yes—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71#832—22no—
72#832—24no—
73#830—21no—
74#824—44no—
75#818—24no—
76#817—24no—
77#810—22no—
78#808—24no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84#533—31no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#498—61no—
90#498—61no—
91#486—61no—
92#481—62no—
93#477—61no—
94#476—62no—
95#474—61no—
96#468—62no—
97#461—61no—
98#460—61no—
99#431—31no—
100#428—62no—
101#423—31no—
102C2×#101—62no—
103C4×M16C2×D311yescharacteristic
104C4×M16C2×D311yescharacteristic
105#329—22no—
106Q8⋊C8—62no—
107C3×#41—82no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—22no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—42no—
121C2×Dic6—42no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C3×M16—24no—
125C3×M16—22no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138#214—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150#208—42no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×#41—21no—
154C22×Q8—61no—
155C2×SD16—62no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—32no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8D1211yescharacteristic
166#127D1211yescharacteristic
167#125—61no—
168#124—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171#121—21no—
172#121—21no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×D4—21no—
176C4×D4—21no—
177#117—61no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—61no—
180C2×C42C22×D311yescharacteristic
181#107—62no—
182#107—64no—
183#106—64no—
184#106—64no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188#103—62no—
189#102—62no—
190#102—64no—
191#101—124no—
192C22⋊C8—61no—
193#97—21no—
194#97#6712yes—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—61no—
198C23×C3—41no—
199C23×C3#12111yescharacteristic
200C3×Q8—81no—
201C3×D4—41no—
202C3×D4—41no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4—82no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—82no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×Dic3—44no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222Dic6—41no—
223Dic6—41no—
224Dic6—42no—
225Dic6—82no—
226Dic6—81no—
227Dic6—81no—
228C3×C8—24no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232#41—82no—
233C2×Q8—21no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—122no—
237C2×Q8—122no—
238C2×Q8—121no—
239C2×Q8—121no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—41no—
244C22×C4—41no—
245C22×C4—62no—
246C22×C4#23511yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249SD16—242no—
250M16—24no—
251M16—22no—
252M16—41no—
253M16—122no—
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—62no—
257C2×C8—62no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—121no—
273C42—121no—
274C42C2×D1211yescharacteristic
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#39211yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—81no—
289C3×C4#45311yescharacteristic
290C3×C4#45411yescharacteristic
291C3×C4#45612yes—
292C3×C4#47112yes—
293Dic3—81no—
294Dic3—82no—
295Dic3—81no—
296C23—41no—
297C23#88411yescharacteristic
298Q8—81no—
299Q8—122no—
300Q8—121no—
301Q8—121no—
302Q8—242no—
303Q8—241no—
304Q8—241no—
305D4—41no—
306D4—42no—
307D4—41no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—124no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#93011yescharacteristic
324C2×C4#93711yescharacteristic
325C2×C4#94011yescharacteristic
326C8—24no—
327C8—42no—
328C8—122no—
329C8—122no—
330C2×C3—41no—
331C2×C3—161no—
332C2×C3#200511yescharacteristic
333C2×C3#200611yescharacteristic
334C2×C3#201911yescharacteristic
335C22—21no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#496011yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—241no—
347C4—242no—
348C4—241no—
349C4#474812yes—
350C4#482211yescharacteristic
351C4#482612yes—
352C4#482711yescharacteristic
353C3#1011611yescharacteristic, complemented, Sylow
354C2—41no—
355C2—161no—
356C2#7009111yescharacteristic
357C2#7038911yescharacteristic
358C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1525C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1525D3
hypocenterC3#10116
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
5. #69877C2semidirect
6. #69856C2semidirect
7. #67669C2semidirect
353. C3#10116semidirect

Maximal subgroups

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

subgroupindex
#833212
#832712
#785062
C3×#15252
#698772
#698562
#676692
#101163