Finite Group Databasebeta

← #10696015

Subgroups — #10696015

Order 768.

534 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2592C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic
3#82448C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic
5#71623C211yesmaximal, characteristic, complemented
6#69493C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10010—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#397—21no—
13C3×#393—21no—
14C3×#391—21no—
15C3×#390—21no—
16C3×#388—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5228—21no—
19#5228—21no—
20#5218—21no—
21#5060—21no—
22#5055—21no—
23#5054—22no—
24#5031—21no—
25#5030—22no—
26#5020—21no—
27#4971—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4540—22no—
32#4535—22no—
33#4523—22no—
34#3215—31no—
35#3113—31no—
36#2592D311yescharacteristic
37#1986—31no—
38#1527—31no—
39#1491—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C42—21no—
60C2×C3×C42D412yes—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63C2×#216—22no—
64C2×#214—22no—
65#926—21no—
66#924—21no—
67#924—21no—
68C2×#208—21no—
69#870—21no—
70#869D411yescharacteristic
71#869D411yescharacteristic
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—21no—
77#835—22no—
78#835—22no—
79#832—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#808—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86#803—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#533—31no—
91C2×#124C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93#497—61no—
94#489—61no—
95#483—61no—
96#475—61no—
97#464—62no—
98#462—61no—
99#451—61no—
100#451—61no—
101#450—61no—
102#441—62no—
103#432—31no—
104#432—31no—
105C4×M16—31no—
106#397—21no—
107#393—21no—
108#391—21no—
109#390—21no—
110#388—22no—
111C4×C4⋊C4C2×D311yescharacteristic
112#339—62no—
113#329—62no—
114Q8⋊C8—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×Dic6—21no—
133C2×Dic6—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—44no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—82no—
145C3×C42—42no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—41no—
153#216—44no—
154#216—82no—
155#214—82no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163#208—22no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C22×Q8#6711yescharacteristic
169C22×Q8#6711yescharacteristic
170C22×D4#6711yescharacteristic
171C2×Q16—62no—
172C2×Q16—62no—
173C2×SD16—62no—
174C2×M16—61no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177#125—61no—
178#124—21no—
179#124—21no—
180#124—21no—
181#124—21no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—61no—
189C4×Q8—61no—
190#117—61no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42#6712yes—
200C2×C42C22×D311yescharacteristic
201#106—62no—
202C4⋊C8—32no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#97—32no—
211#97—61no—
212#97—61no—
213C4×C8—61no—
214#95—22no—
215#95—22no—
216#95—21no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3#12011yescharacteristic
223C3×Q8—22no—
224C3×Q8—22no—
225C3×Q8—41no—
226C3×Q8—41no—
227C3×Q8—42no—
228C3×Q8—81no—
229C3×D4—42no—
230C3×D4—81no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×Dic3—41no—
253C2×Dic3—41no—
254Dic6—42no—
255Dic6—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C24—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—42no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4C2×#6711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278Q16—124no—
279Q16—242no—
280SD16—242no—
281M16—62no—
282M16—121no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—82no—
303C22⋊C4—121no—
304C42—42no—
305C42—41no—
306C42—41no—
307C42—121no—
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#39211yescharacteristic
320C22×C3#45511yescharacteristic
321C22×C3#45611yescharacteristic
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—82no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4#45512yes—
332Dic3—81no—
333Dic3—81no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#95311yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—42no—
345Q8—81no—
346Q8—122no—
347Q8—122no—
348D4—42no—
349D4—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#93711yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93811yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#94011yescharacteristic
373C8—122no—
374C8—122no—
375C8—122no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—161no—
379C2×C3#200411yescharacteristic
380C2×C3#200711yescharacteristic
381C2×C3#201011yescharacteristic
382C22—21no—
383C22—21no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#522311yescharacteristic
390C22#524211yescharacteristic
391C22#524611yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—241no—
402C4—241no—
403C4#506112yes—
404C3#1001011yescharacteristic, complemented, Sylow
405C2—21no—
406C2—21no—
407C2—161no—
408C2#7171511yescharacteristic
409C2#7173111yescharacteristic
410C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2592C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2592D3
hypocenterC3#10010
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2592

Complements

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

NH ≅ G/Ntype
3. #82448C2semidirect
5. #71623C2semidirect
6. #69493C2semidirect
7. #67621C2semidirect
404. C3#10010semidirect

Maximal subgroups

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

subgroupindex
C3×#25922
#829492
#824482
#717182
#716232
#694932
#676212
#100103