Finite Group Databasebeta

← #10697400

Subgroups — #10697400

Order 768.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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#83308C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic, complemented
3#79065C211yesmaximal, characteristic, complemented
4C3×#1963C211yesmaximal, characteristic, complemented
5#69433C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic, complemented
7#67716C211yesmaximal, characteristic, complemented
8#10214—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#104—22no—
14C3×#376C2211yescharacteristic
15#5134—22no—
16#5089—21no—
17#5081—21no—
18#5074—21no—
19#5071—21no—
20#5037—22no—
21#5035—22no—
22#5019—21no—
23#5015—21no—
24#5008—21no—
25#5007—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4850—21no—
29#4848—21no—
30#4846—21no—
31#4841—21no—
32#4452—22no—
33#4451—22no—
34#3254—31no—
35#3235—31no—
36#3214—31no—
37#1963D311yescharacteristic, complemented
38#1506—31no—
39#1505—31no—
40#1307—31no—
41C2×C3×#41—21no—
42C2×#235—21no—
43C2×C3×M16D412yes—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#204—22no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#104—44no—
74C3×#95—21no—
75C3×#95—21no—
76#832—22no—
77#832—22no—
78#830D412yes—
79#830D412yes—
80#821—22no—
81#818—22no—
82#816—22no—
83#810—22no—
84#809—22no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#803—21no—
91#533—31no—
92#532—31no—
93#532C2×D311yescharacteristic
94#501—61no—
95#496—61no—
96#490—61no—
97#489—61no—
98#482—62no—
99#477—61no—
100#476—61no—
101#474—61no—
102#474—61no—
103#468—62no—
104#461—61no—
105#461—61no—
106#460—61no—
107#459—61no—
108#449—62no—
109#432—31no—
110#432—31no—
111#423C2×D311yescharacteristic
112C2×#104—22no—
113#376C2×D311yescharacteristic
114#328—62no—
115#327—62no—
116C3×#41—42no—
117C3×#41—42no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—41no—
129#235—82no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C3×M16—42no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—22no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—41no—
150#214—82no—
151#213—82no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165#204—82no—
166C2×#41—21no—
167C2×#41—61no—
168C2×SD16—62no—
169C2×SD16—62no—
170C2×D8—62no—
171C2×M16#6712yes—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8D1211yescharacteristic
176#127—31no—
177#127D1211yescharacteristic
178#125—61no—
179#124—21no—
180#124—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183#121—21no—
184#121—21no—
185#121—61no—
186#121—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—61no—
192C4×D4—61no—
193#117—61no—
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#107—62no—
198#106—62no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203#104—44no—
204#103—62no—
205#103—62no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—32no—
211C22⋊C8—32no—
212#97—61no—
213#97—61no—
214#95—21no—
215#95—21no—
216C23×C3—41no—
217C23×C3#12711yescharacteristic
218C22×D3—41no—
219C3×Q8—42no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—82no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241#67—82no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245D12—42no—
246D12—41no—
247D12—41no—
248D12—81no—
249C4×D3—82no—
250Dic6—42no—
251Dic6—41no—
252Dic6—41no—
253Dic6—81no—
254C3×C8—42no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257#41—42no—
258#41—42no—
259#41—242no—
260C2×Q8—21no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—121no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278SD16—242no—
279SD16—242no—
280D8—242no—
281M16—42no—
282C2×C8—22no—
283C2×C8—62no—
284C2×C8—62no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—22no—
301C42—41no—
302C42—121no—
303C42C2×D1211yescharacteristic
304C42#23511yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—82no—
323C3×C4—81no—
324C3×C4#45311yescharacteristic
325C3×C4#45411yescharacteristic
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329C23—41no—
330C23—121no—
331C23#94011yescharacteristic
332Q8—42no—
333Q8—122no—
334Q8—121no—
335Q8—121no—
336Q8—241no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—82no—
342D4—122no—
343D4—121no—
344D4—121no—
345D4—241no—
346D4—242no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—242no—
361C2×C4#88211yescharacteristic
362C2×C4#88211yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4#93011yescharacteristic
366C2×C4#93711yescharacteristic
367C8—42no—
368C8—122no—
369C8—122no—
370C2×C3—41no—
371C2×C3—161no—
372C2×C3#200611yescharacteristic
373C2×C3#200811yescharacteristic
374C2×C3#200911yescharacteristic
375D3—161no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#496011yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—81no—
393C4—241no—
394C4—241no—
395C4—241no—
396C4#506211yescharacteristic
397C4#507711yescharacteristic
398C3#1021411yescharacteristic, complemented, Sylow
399C2—41no—
400C2—161no—
401C2—481no—
402C2#6966411yescharacteristic
403C2#7040911yescharacteristic
404C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1963C2
socleC22×C3#392
prime radical#83308C2
hypercenter#1963D3
hypocenterC3#10214
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #83274C2semidirect
3. #79065C2semidirect
4. C3×#1963C2semidirect
5. #69433C2semidirect
6. #69432C2semidirect
7. #67716C2semidirect
37. #1963D3semidirect
398. C3#10214semidirect

Maximal subgroups

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

subgroupindex
#833082
#832742
#790652
C3×#19632
#694332
#694322
#677162
#102143